Lectures on
Topics In One-Parameter Bifurcation Problems
By
P. Rabier
Tata Institute of Fundamental Research
Bombay
1985
Lectures on
Topics In One-Parameter Bifurcation Problems
By
P. Rabier
Lectures delivered at the
Indian Institute of Science, Bangalore
under the
T.I.F.R.–I.I.Sc. Programme In Applications of
Mathematics
Notes by
G. Veerappa Gowda
Published for the
Tata Institute of Fundamental Research
Springer-Verlag
Berlin Heidelberg New York Tokyo
1985
Author
P. Rabier
Analyse Numérique, Tour 55-65, 5e étage
Université Pierre et Marie Curie
4, Place Jussieu
75230 Paris Cedex 05
France
c
Tata Institute of Fundamental Research, 1985
ISBN 3-540-13907-9 Springer-Verlag, Berlin. Heidelberg.
New York. Tokyo
ISBN 0-387-13907-9 Springer-Verla, New York. Heidelberg.
Berlin. Tokyo
No part of this book may be reproduced in any
form by print, microfilm or any other means without written permission from the Tata Institute of
Fundamental Research, Colaba, Bombay 400 005
Printed by M. N. Joshi at The Book Centre Limited,
Sion East, Bombay 400 022 and published by H. Goetze,
Springer-Verlag, Heidelberg, West Germany
Printed In India
Preface
This set of lectures is intended to give a somewhat synthetic exposition
for the study of one-parameter bifurcation problems. By this, we mean
the analysis of the structure of their set of solutions through the same
type of general arguments in various situations.
Chapter I includes an introduction to one-parameter bifurcation
problems motivated by the example of linear eiqenvalue problems and
step by step generalizations lead to the suitable mathematical form. The
Lyapunov-Schmidt reduction is detailed next and the chapter is completed by an introduction to the mathematical method of resolution,
based on the Implicit function theorem and the Morse lemma in the simplest cases. The result by Crandall and Rabinowitz [7] about bifurcation
from the trivial branch at simple characteristic values is given as an
example for application.
Chapter II presents a generalization of the Morse lemma in its
“weak” form to mappings from Rn+1 into Rn . A slight improvement
of one degree of regularity of the curves as it can be found in the literature, is proved, which allows one to include the case when the Implicit
function theorem applies and is therefore important for the homogeneity
of the exposition. The relationship with stronger versions of the Morse
lemma is given for the sake of completeness but will not be used in the
sequel.
Chapter III shows how to apply the results of Chapter II to the study
of one-parameter bifurcation problems. Attention is confined to two
general examples. The first one deals with problems of bifurcation from
the trivial branch at a multiple characteristic value. A direct application
v
vi
Preface
may be possible but, for higher non-linearities, a preliminary change
of scale is necessary. The justification of this change of scale is given
at an intuitive level only, because a detailed mathematical justification
involves long and tedious technicalities which do not help much for understandig the basic phenomena, even if they eventually provide a satisfactory justification for the use of Newton diagrams (which we do not
use however). The conclusions we draw are, with various additional information, those of McLeod and Sattinger [23]. The second example
is concerned with a problem in which no particular branch of solutions
is known a priori. It is pointed out that while the case of a simple singularity is without bifurcation, bifurcation does occur in general when
the singularity is multiple. Also, it is shown how to get further details
on the location of the curves when the results of Chapter II apply after
a suitable change of scale and how this leads at once to the distinction
between “turning points” and “hysteresis points” when the singularity is
simple.
Chapter IV breaks with the traditional exposition of the LyapunovSchmidt method, of little and hazardous practical use, because its assumed data are not known in the applications while the imperfection
sensitivity of the method has not been evaluated (to the best of our
knowledge at least). Instead, we present a new, more general (and we
believe, more realistic) method, introduced in Rabier-El Hajji [33] and
derived from the “almost” constructive proofs of Chapter II. Optimal
rate of convergence is obtained. For the sake of brevity, the technicalities of § 5 have been skipped but the first four sections fully develop all
the main ideas.
Chapter V introduces a new method in the study of bifurcation problems in which the nondegeneracy condition of Chapter II is not fulfilled.
Actually, the method is new in that it is applied in this context but similar techniques are classical in the desingularization of algebraic curves.
We show how to find the local zero set of a C ∞ real-valued function
of two variables (though the regularity assumption can be weakened in
most of the cases) verifying f (0) = 0, D f (0) = 0, D2 f (0) , 0 but
det D2 f (0) = 0 (so that the Morse condition fails). This method is applied to a problem of bifurcation from the trivial branch at a geometri-
Preface
vii
cally simple characteristic value when the nondegeneracy condition of
Crandall and Rabinowitz is not fulfilled (i.e. the algebraic multiplicity
as > 1). The role played by the generalized null space is made clear
and the result complements Krasnoselskii’s bifurcation theorem in the
particular case under consideration.
Preface
viii
Acknowledgement
I wish to thank Professor M.S. NARASIMHAN for inviting me and
giving me the opportunity to deliver these lectures at the Tata Institute
of Fundamental Research Centre, Bangalore, in July and August 1984.
I am also grateful to Drs. S. KESAVAN and M. VANNINATHAN who
initially suggested my visit.
These notes owe much to Professor S. RAGHAVAN and Dr. S. KESAVAN who used a lot of their own time reading the manuscript. They
are responsible for numerous improvements in style and I am more than
thankful to them for their great help.
Contents
Preface
v
1 Introduction to One-Parameter Bifurcation Problems
1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Lyapunov-Schmidt Reduction . . . . . . . . . . . . 10
1.3 Introduction to the Mathematical Method...... . . . . . . 15
2 A generalization of the Morse Lemma
2.1 A Nondegeneracy Condition For......... . . . . .
2.2 Practical Verification of the Condition (R-N.D.).
2.3 A Generalization of the Morse Lemma..... . . .
2.4 Further Regularity Results. . . . . . . . . . . .
2.5 A Generalization of the Strong Version..... . . .
.
.
.
.
.
29
29
33
40
48
54
3 Applications to Some Nondegenerate problems
3.1 Equivalence of Two Lyapunov-Schmidt Reductions. . . .
3.2 Application to Problems of Bifurction..... . . . . . . . .
3.3 Application to a Problem...... . . . . . . . . . . . . . . .
57
58
65
85
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
4 An Algorithm for the Computation of the Branches
4.1 A Short Review of the Method of Chapter II. . . . . .
4.2 Equivalence of Each Equation with a... . . . . . . . . .
4.3 Convergence of the Successive Approximation Method.
4.4 Description of the Algorithm. . . . . . . . . . . . . . .
4.5 A Generalization to the Case..... . . . . . . . . . . . .
ix
.
.
.
.
.
99
100
101
106
108
111
Contents
x
4.6
5
Application to One-Parameter Problems. . . . . . . . . . 116
Introduction to a Desingularization Process....
5.1 Formulation of the Problem and Preliminaries. . . . . . .
5.2 Desingularization by Blowing-up Procedure. . . . . . . .
5.3 Solution Through the Implict Function.... . . . . . . . .
5.4 Solution Through the Morse Lemma.... . . . . . . . . . .
5.5 Iteration of the Process. . . . . . . . . . . . . . . . . . .
5.6 Partial Results on the Intrinsic Character of the Process .
5.7 An Analytic Proof of Krasnoselskii’s Theorem.... . . . .
5.8 The Case of an Infinite Process. . . . . . . . . . . . . .
5.9 Concluding Remarks on Further Possible Developments.
125
127
130
135
139
146
155
161
170
181
Appendix 1: Practical Verification of the Conditions.....
185
Appendix 2: Complements of Chapter IV
193
Chapter 1
Introduction to
One-Parameter Bifurcation
Problems
1.1 Introduction
In this, section, we introduce one-parameter bifurcation problems 1
through the example of linear eigenvalue problems. In increasing order of generality, they first lead to non-linear eigenvalue problems, next,
to problems of bifurction from the trivial branch and finally to a large
class of problems for which a general mathematical analysis can be developed.
1.1a Linear Eigenvalue Problems
Let X be a real vector space. Given a linear operator L : X → X, we
consider the problem of finding the pairs (λ, x)ǫR × X that
x = λLx.
For λ = 0, x = 0 is the unique solution. For λ , 0, and setting
τ = 1/λ, it is equivalent to
τx = Lx.
1
2
2
1. Introduction to One-Parameter Bifurcation Problems
The values τǫR such that there exists x , 0 satisfying the above
equation are called eigen values of L. When λ , 0, the corresponding
value λ = 1/τ is called a characteristic value of L.
It may happen that every non-zero real number λ is a characteristic
value of L. For instance, if X = D ′ (R) (distributions over R) and L is
the operator Lx = x′′ ǫD(R). Then
√
x = λx′′ ↔ x(s) = es/ λ .
But this is not the case in general. In what follows, we shall assume
that X is a real Banach space and that LǫL (x) is compact.
From the special theory of linear operators in Banach spaces, it is
known that
(i) The characteristic values of L form a sequence {λ j } j≥1 with no
cluster point (the sequence is finite if dim X < ∞).
(ii) For every λǫR, Range (I − λL) is closed and dim Ker(I − λL) =
codim Range (I − λL) < ∞ (and is greater than or equal to 1 if
and only if λ = λ j for some j).
Let us now set
H(λ, x) = x − λLx
so that the problem consists in finding the pairs (λ, x)ǫR × X such that
H(λ, x) = 0. The set of solutions of this equation (zero set of H) is the
S
union of the line {(λ, 0); λǫR} (trivial branch) and the set {λ j } × E j
j≥1
3
where E j denotes the eigenspace associated with the characteristic value
λ j.
Now, let us take a look at the local structure of the zero set of H
around a given point (λ0 , 0), λ0 ǫR ; If λ0 is not a characteristic value of
L, it is made up of exactly one curve (the trivial branch itself). If λ0 = λ j
for some j ≧ 1, the structure changes, since there are solutions of the
form (λ j , x), xǫE j , arbitrarily close to (λ j , 0). The existence of nontrivial
solutions (i.e. which do not belong to the trivial branch) around a point
(λ j , 0) is referred to as a bifurcation phenomenon (here, form the trivial
branch) and the points (λ j , 0) are called bifurction points. Bifurcation
can be viewed as a breaking of smoothess of the local zero set whereas
data whereas all the data are smooth.
1.1. Introduction
3
1.1b Generalization I: Problems of bifurcation from the trivial branch.
A natural extension is when the linear operator L is replaced by a mapping T : X → X (nonlinear in general) such that T (0) = 0. The problem
becomes: Find (λ, x)ǫR × X such that
x = λT (x).
Again, the pairs (λ, 0)λǫR are always solutions of this equation (trivial branch). On the basis of the linear case, a natural question is to know
whether there are “bifurcation points” on the trivial branch, namely solutions (λ◦ , 0) around which nontrivial solutions always exist.
Theorem 1.1 (Necessary condition). Assume that T is differentiable at 4
the origin and the linear operator DT (0)ǫL (X) is compact. Then a
necessary condition for (λ0 , 0) to be a bifurcation point of the equation
x = λT (x) is that λ0 is a characteristic value of DT (0).
Proof. Write
and let
(λ(i) , x(i) )
T (x) = DT (0) · x + ◦(||x||)
be a sequence tending to (λ0 , 0) with x(i) , 0 and
x(i) = λ(i) T (x(i) ).
Thus,
x(i) = λ(i) DT (0) · x(i) + 0(||x(i) ||)
Dividing by ||x(i) || , 0, we get
x(i)
x(i)
(i)
=
λ
DT
(0).
+ 0(1)
||x(i) ||
||x(i) ||
(i) The sequence ||xx(i) || is bounded. Due to the compactness of the
operator DT (0), we may assume, after considering a subsequence, that
the right hand
(i) side tends to a limit v, which is then the limit of the
sequence ||xx(i) || as well. Of course, v , 0 and making i tend to +∞, we
find
v = λ0 DT (0) · v,
which shows that λ0 is a characteristic value of DT (0).
4
5
1. Introduction to One-Parameter Bifurcation Problems
Remark 1.1. As T is nonlinear (a very general assumption !) it is impossible, without additional hypotheses to expect more than local results
(in contrast to the linear case where global results are obtained).
Remark 1.2. Even when λ0 is a characteristic value of DT (0), bifurcation is not ensured. For instance, take X = R2 and x = (x1 , x2 ) with


 x1 + x32 
T (x1 , x2 ) = 
 .
x2 − x31
Here, DT (0) = I , whose unique characteristic value is λ1 = 1. The
equation x = λT (x) becomes
x1 = λx1 + λx32 ,
x2 = λx2 − λx31 .
Multiplying the first equation by x2 and the second one by −x1 and
adding the two we get λ(x41 + x42 ) = 0. Hence for λ around 1, we must
have x = 0 and no bifurcation occurs.
These nonlinear eigenvalue problems are particular cases of a more
general class called problems of bifurcation from the trivial branch. By
definition, a problem of bifurcation from the trivial branch is an equation
of the form
x = λLx − φ(λ, x),
6
where LǫL (X) and φ is a nonlinear operator from X to itself satisfying
φ(λ, 0) = 0 for λǫR,
(1.1)
φ(λ, x) = 0(||x||),
(1.2)
for x around the origin, uniformly with respect to λ on bounded intervals.
It is equivalent to saying that a problem of bifurcation from the trivial
branch consists in finding the zero set of the mapping
H(λ, x) = x − λLx + φ(λ, x)
From our assumptions, the pairs (λ, 0), λǫR are all in the zero set of
H (trivial branch). Note, however, that our definition does not include
1.1. Introduction
5
all mappings having the trivial branch in their zero set. Two reasons
motivate our definition. First, problems of this type are common in the
literature, for they correspond to many physical examples. Secondly,
from a mathematical stand-point, their properties allow us to make a
general study of their zero set. In particular, when L is compact, a proof
similar to that of theorem 1.1 shows that λ0 needs to be characteristic
value of L for (λ0 , 0) to be a bifurcation point. Actually, by writing the
Taylor expansion of T about the origin
T (x) = DT (0) · x + R(x),
with R(x) = 0(||x||), it is clear that nonlinear eigenvalue problems are
a particular case of problems of bifurcation from the trivial branch in
which L = DT (0) and φ(λ, x) = −λR(x).
The most famous result about problems of bifurction from the trivial 7
branch is a partial converse of Theorem 1.1 due to Kranoselskii.
Theorem 1.2 (Krasnoselskii). Assume that L is compact and λ0 is a
characteristic value of L with odd algebraic multiplicity, Then (λ0 , 0) is
a bifurcation point (i.e. there are solutions (λ, x)ǫR×X −{0} of H(λ, x) =
0 arbitrarily close to (λ0 , 0)).
The proof of Theorem 1.2 is based on topological degree arguments
and will not be given here (cf. [19], [27]). It is a very general result but
it does not provide any information on the structure of the zero set of H
near (λ0 , 0), a question we shall be essentially interested in, throughout
these notes.
COMMENT 1.1. (Algebraic and geometric multiplicity of a characteristic value): Let L be compact. Given a characteristic value λ0 of L,
it is well-known that the space Ker(I − λ0 L) is finite dimensional. Its
dimension is called the geometric multiplicity of λ0 .
The spectral theory of compact operators in Banach spaces (see e.g.
[9]) provides us with additional information; namely, there is an integer
r ≥ 1 such that
(i) dim Ker(I − λ0 L)r < ∞,
6
1. Introduction to One-Parameter Bifurcation Problems
(ii) X = Ker(I − λ0 L)r ⊕ Range(I − λ0 L)r .
8
In addition, r is characterized by
′
Ker(I − λ0 L)r = Ker(I − λ0 L)r for every r′ ≥ r.
The dimension of the space Ker(I − λ0 L)r is called the algebraic
multiplicity of λ0 . The algebraic multiplicity of λ0 is always greater
holds if an only if r = 1. If so, it follows from property (ii) that
X = Ker(I − λ0 L) ⊕ Range(I − λ0 L).
A typical example of this situation is when X is a Hilbert space and
L is self-adjoint.
COMMENT 1.2. In particular, Krasnoselskii’s theorem applied when
r = 1 and dim Ker(I − λ0 L) = 1. For instance, this happens when L
is the inverse of a second order elliptic linear operator associated with
suitable boundary conditions and λ0 is the “first” characteristic value of
L; this result is strongly related to the maximum principle through the
Krein-Rutman theorem. (See e. g. [20, 35]).
COMMENT 1.3. For future use, note that the mapping φ is differentiable with respect to the x variable at the origin with
D x φ(λ, 0) = 0 for every λǫR,
(1.3)
as it follows from (1.2).
9
EXAMPLES. The most important examples of problems of bifurcation
from the trivial branch came from nonlinear partial differential equations. For instance, let us consider the model problem



−△u + λu ± uk = 0 in Ω,


uǫH 1 (Ω),
0
where Ω is a bounded open subset of RN and k ≥ 2 is an integer.
1.1. Introduction
7
Under general assumptions on the boundary ∂Ω of Ω, it follows
from the Sobolev embedding theorems that uk ǫH −1 (Ω) for k ≤ N+2
N−2 if
N > 2, any 1 ≤ k < +∞ for N = 1 and N = 2.
Denoting by LǫL (H −1 (Ω), H01 (Ω)) the inverse of the operatoe −△,
the problem becomes equivalent to



u − λLu + L(uk ) = 0,


uǫH 1 (Ω).
0
Note that the restriction of L to the space H01 (Ω) is compact and the
mapping
uǫH01 (Ω) → Luk ǫH01 (Ω)
is of class C ∞ . In this example, the nonlinearity uk can be replaced
by F(u) (respectively F(λ, u)) where F(u)(x) = f (x, u(x)) (respectively
F(λ, u)(x) = f (λ, x, u(x))) and f is a Carathéodory function satisfying
some suitable growth conditions with respect to the second (respectively
third) variable (see eg. Krasnoselskii [19], Rabier [30]).
Another example with a non-local nonlinearity is given by the von 10
Karman equations for the study of the buckling of thin plates. The problem reads: Find u such that



u − λLu + C(u) = 0,


uǫH 2 (ω),
0
where ω is an open bounded subset of the plane R2 , LǫL (H02 (ω)) is
compact and C is a “cubic” nonlinear operator. The operator L takes
into account the distribution of lateral forces along the boundary ∂ω, the
intensity of these forces being proportional to the scalar λ. For “small”
values of λ, the only solution is u = 0 but, beyond a certain critical value,
nonzero solutions appear: this corresponds to the (physically observed)
fact that the plate jumps out of its for sufficiently “large” λ (see e. g.
Berger [1] Ciarlet-Rabier [6]).
Coming back to the general case, our aim is to give as precise a
description as possible of the sero set of H around the point (λ0 , 0).
Assuming L is compact, we already know the answer when λ0 is not a
1. Introduction to One-Parameter Bifurcation Problems
8
characteristic value of L: the zero set coincides with the trivial branch.
In any case, it is convenient to shift the origin and set
11
λ = λ0 + µ
(1.4)
Γ(µ, x) = φ(λ0 + µ, x)
(1.5)
so that the problem amounts to finding the zero set around the origin
(abbreviated as local zero set) of the mapping
G(µ, x) = x − (λ0 + µ)Lx + Γ(µ, x)
(1.6)
Note that the mapping Γ verifies the properties
Γ(µ, 0) = 0 for every µǫR,
(1.7)
Γ(µ, x) = 0(||x||),
(1.8)
around the origin, uniformly with respect to µ on bounded intervals. In
particular, Γ is differentiable with respect to the x-variable and
D x Γ(µ, 0) = 0 for every µǫR.
(1.9)
1.1c GENERALIZATION II :
Let us consider a problem of bifurcation from the trivial branch with
compact operator LǫL (X), put under the form G(µ, x) = 0 after fixing
the real number λ0 as described above. From (1.7)
Dµ Γ(0) = 0.
(1.10)
Together with (1.9), we see that the (global) derivative DG(0) is the
mapping
(µ, x)ǫR × X → (I − λ0 L)xǫX.
(1.11)
Hence
Ker DG(0) = R × Ker(I − λ0 L),
RangeDG(0) = Range(I − λ0 L).
(1.12)
(1.13)
1.1. Introduction
12
9
As a result, Range Dg(0) is closed and dim Ker DG(0) = codim
RangeDG(0) + 1 < ∞ (≥ 1, if and only if λ0 is a characteristic value of
L). In other words, DG(0) is a Fredholm operator with index 1. Recall
e to a Banach space Y is
that a linear operator A from a Banach space X
said to be a Fredholm operator if
(i) Range A is closed,
(ii) dim ker A < ∞, codim Range A < ∞.
In this case, the difference
dim Ker A − codimRangeA
is called the index of A. For the definition and further properties of
Fredholm operators, see Kato [17] or Schecter [36].
Remark 1.3. One should relate the fact that the index of DG(0) is 1 to
the fact that the parameter µ is one-dimensional. This will be made more
clear in Remark 2.2 later.
Although the parameter µ has often a physical significance (end
hence must be distinguished from the variable x for physical reasons), it
is not always desirable to let it play a particular role in the mathematical
approach of the problem.
The suitable general mathematical framework is as follows:
e and Y and let G : X
e→
Let there be given two real Banach spaces X
Y be a mapping satisfying the conditions
13
G(0) = 0
(1.14)
G is differentiable at the origin ,
(1.15)
DG(0) is a Fredholm operator with index 1.
(1.16)
Naturally, G need not be defined everywhere, but in a neighbourhood of the origin only. However, for notational convenience, we shall
repeatedly make such an abuse of notation in the future, without further
mention.
10
1. Introduction to One-Parameter Bifurcation Problems
In our previous example of problems of bifurcation from the trivial
e = R × X, Y = X and G(µ, 0) = 0 for µǫR. None of
branch, one has X
these assumptions is required here. In particular, nothing ensures that
the trivial branch is in the local zero set of G. As a matter of fact, no
branch of solutions (trivial or not) is supposed to be known a priori.
The first step of the study consists in performing the LyapunovSchmidt reduction allowing us to reduce the problem to a finite dimensional one and this will be done in the next section.
1.2 The Lyapunov-Schmidt Reduction
e and Y be two real Banach spaces and G : X
e → Y a mapping of
Let X
m
class C , m ≥ 1 verifying (1.14) - (1.16). Let us set
e1 = Ker DG(0),
X
Y2 = RangeDG(0).
(2.1)
(2.2)
14
e1 has finite
By hypothesis, Y2 has finite codimension n ≥ 0 as X
e2 and Y1 be two topological complements of X
e1
dimension n + 1. Let X
and Y respectively.
Remark 2.1. (Existence of topological complements) From the Hahne has a topological
Banach theorem, each one-dimensional subspace of X
e (direct sum
complement. Then, each finite dimensional subspace of X
of a finite number of one-dimensional subspaces) has a topological complement (the intersection of the complements of these one-dimensional
e1 has a topological complement. Next, the
subspaces). In particular, X
existence of a topological complement of Y2 is due to the fact that Y2
is closed with finite codimension. Any (finite-dimensional) algebraic
complement of Y2 is closed and hence is also a topological complement.
Details are given for instance, in Brezis [4].
Thus we can write
e= X
e1 ⊕ X
e2
X
(2.3)
1.2. The Lyapunov-Schmidt Reduction
Y = Y1 ⊕ Y2 .
11
(2.4)
Let Q1 and Q2 denote the (continuous) projection operators onto Y1
e set
and Y2 respectively. On the other hand, for every e
xǫ X,
e1 , e
e2 .
e
x=e
x1 + e
x2 , e
x1 ǫ X
x2 ǫ X
With this notation, the equation G(e
x) = 0 goes over into the system
Q1 G(e
x1 + e
x2 ) = 0ǫY1 ,
Q2 G(e
x1 + e
x2 ) = 0ǫY2 ,
(2.5)
(2.6)
15
Now, from our assumptions, one has
e2 , Y2 ).
DG(0)|Xe2 ǫI som(X
(2.7)
e2 and
Indeed, DG(0)|Xe2 is clearly one-to-one, onto (by definition of X
Y2 ) and continuous. As Y2 is closed in Y, it is a Banach space by itself
and the result follows from the open mapping theorem. Thus equation
(2.6) is solved in a neighbourhood of the origin by
e
x2 = e
ϕ(e
x1 )
e
ϕ(0) = 0
e1 → X
e2 is a uniquely determined C m mapping (Implicit
where e
ϕ : X
function theorem). After substituting in the first equation, we find the
reduced equation
Q1 G(e
x1 + e
ϕ(e
x1 )) = 0ǫY1 ,
(2.8)
e G(e
equivalent to the original equation : e
xǫ X,
x) = 0, around the origin.
From now on, we drop the index “1” in the notation of the generic elee1 . The mapping
ment of the space X
e
xǫX1 → f (e
x) = Q1G(e
x+e
ϕ(e
x))ǫY1 ,
(2.9)
12
1. Introduction to One-Parameter Bifurcation Problems
whose local zero set is made up of the solution of the reduced equation
(2.8) is called the reduced mapping (note of course that f verifies f (0) =
0).
Therefore, we have reduced the problem of finding the local zero 16
set of G to finding the local zero set of the reduced mapping f in (2.9),
which is of class C m from a neighbouhood of the origina in the (n + 1)e1 into the n-dimensional space Y1 .
dimensional spcae X
Remark 2.2. More generally, assume DG(0) is a Fredhlom operator
with index p ≥ 1. The same process works; we end up with a reduced
e1 ((n + p)-dimensional) into the space Y1 (nmappinhy from the space X
dimensional), so that p can be thought of as the number of “free” real
variables (cf. Remark 1.3).
Two Simple Properties.
The derivative at the origin of the reduced mapping f(cf. (2.9)) is
immediately found to be
Q1 DG(0)(Iex1 + De
ϕ(0)).
But Q1 DG(0) = 0, by the definition of the space Y1 , so that
D f (0) = 0
(2.10)
Next, from the characterization of the function e
ϕ and by implict differentiation, we get
ϕ(0)) = 0.
Q2 DG(0)(IXe1 + De
In other words, since Q2 DG(0) = DG(0) by a definition of the space
e1 = Ker DG(0),
Y2 and since X
17
DG(0) · De
ϕ(0) = −DG(0) · IXe1 = 0
(2.11)
On the other hand, the function e
ϕ takes its values in the space so that
e1 , X
e2 ),
De
ϕ(0)ǫL (X
1.2. The Lyapunov-Schmidt Reduction
13
and the relation (2.11) can be written as
Thus, form (2.7)
DG(0)|Xe2 · De
ϕ(0) = 0.
De
ϕ(0) = 0.
(2.12)
The Lyapunov-Schmidt reduction in the case of problems of bifurcation from the trivial branch.
In our examples later, we shall consider the case of problems of
bifurcation from the trivial branch. This is the reason why we are going
to examine the form taken by the Lyapunov-Schmidt reduction in this
context. Of course, this is simply a particular case of the general method
previously described.
Let X = Y and consider a problem of bifurcation from the trivial
branch in the form G(µ, x) = 0 after fixing the real number λ0 . As we
observed earlier,
e1 = Ker DG(0) = R × Ker(I − λ0 L),
X
(2.13)
Y2 = RangeDG(0) = Range(I − λ0 L).
(2.14)
X1 = Ker(I − λ0 L),
(2.15)
Setting
this becomes
18
e1 = R × X1 ,
X
(2.16)
e1 can be identified with a pair (µ, x1 )ǫR × X1 .
so that any element e
x1 ǫ X
Thus given a topological complement X2 of X1 in the space X, we can
make the choice
e2 = {0} × X2 .
X
(2.17)
e1 in R × X is of the form (2.17).
Note that not every complement of X
Nevertheless, such a choice is “standard” in the literature devoted to
problems of bifurcation from the trivial branch. Writing each element
xǫX as a sum
x = x1 + x2 , xǫX1 , x2 ǫX2 ,
1. Introduction to One-Parameter Bifurcation Problems
14
the mapping e
ϕ identifies with a mapping ϕ(= ϕ(µ, x1 )) with values in
X2 , characteried by the relation
Q2 (I − (λ0 + µ)L)(x1 + ϕ(µ, x1 )) + Q2 Γ(µ, x1 + ϕ(µ, x1 )) = 0. (2.18)
Remark 2.3. In particular, ϕ(µ, 0) = 0 since ϕ(µ, 0) is uniquely determined by the relation (2.18) which is satisfied with the choice x1 =
0, ϕ(µ, 0) = 0.
The reduced mapping thus takes the form
f (µ, x1 ) = Q1G(µ, x1 + ϕ(µ, x1 )),
19
where the mapping G is given by (1.6).
As
Q1 (I − λ0 L) = 0,
(2.19)
(2.20)
by definition of the space Y1 , we find
Q1G(µ, x) = −µQ1 Lx + Q1 Γ(µ, x).
But the relation (2.20) also shows that
Q1 L =
and hence
Q1G(µ, x) = −
1
Q1
λ0
(2.21)
µ
Q1 x + Q1 Γ(µ, x).
λ0
Therefore, droppping the index “1” in the notation of the generic element of the space X1 , the reduced mapping takes the form
(µ, x)ǫR × X1 → f (µ, x) = −
µ
µ
Q1 × − Q1 ϕ
λ0
λ0
(µ, x) + Q1 Γ(µ, x + ϕ(µ, x))ǫY1 .
(2.22)
1.3. Introduction to the Mathematical Method......
15
1.3 Introduction to the Mathematical Method of
Resolution (Implicit Function Theorem and
Morse Lemma).
Since f (0) = 0, the first natural tool we can think of, for finding the local
zero set of f , is the Implict function theorem. But we already saw that
D f (0) = 0. Hence the Implict function theorem can be applied when
Y1 = {0} (i.e., n = 0) only. In other words, one must have Y2 = Y so
that DG(0) is onto. If so, the reduced mapping f vanishes identically
and the problem has actually already been solved while performing the
Lyapunov-Schmidt reduction: the local zero set of G is given by the
graph of the mapping e
ϕ, that is to say, the curve
e1 → e
e
e
xǫ X
x+e
ϕ(e
x)ǫ X.
20
The reader would have noticed that since DG(0) is onto, the Lyapunov - Schmidt reduction amounts to applying the Implict function
theorem to the original problem. There is more to say about this apparently obvious situation. Let us come back to the case when the parameter µ is explicitly mentioned in the expression for G (for physical
e = R × X and G = G(µ, x). Then, for
reasons for instance), namely X
every (µ, x)ǫR × X, we have
DG(0) · (µ, x) = µDµG(0) + D xG(0) · x
(3.1)
and there are two possibilities for DG(0) to be onto; either
D xG(0) is onto ,
(3.2)
or
codim Range D xG(0) = 1 and DµG(0) < Range D xG(0).
We shall
X1 = Ker D xG(0).
(3.3)
16
1. Introduction to One-Parameter Bifurcation Problems
When condition (3.2) is fulfilled, there, is an element ξǫX such that
D xG(0) · ξ = −DµG(0).
Hence, DG(0) is the linear mapping
(µ, x)ǫR × X → D xG(0) · (x − µξ)ǫY
21
and it follows that
e1 = Ker DG(0) = {(µ, x)ǫR × X, x − µξǫX1 } =
X
= {(µ, x)ǫR × X, x = µξ + x1 , x1 ǫX1 } =
= Rµ (1, ξ) ⊕ ({0} × X1 ),
e1 = 1,
where Rµ denotes the real line with generic variable µ. As dim X
we must have X1 = {0} (i.e. Ker D xG(0) = {0}) and D xG(0) is then an
isomorphism (in particular, ξ is unique). Thus
e1 = Rµ (1, ξ),
X
a relation showing that the local zero set of G is parametrized by µ (See
Figure 3.1 below)
Figure 3.1: “regular point”
In this case, the origin is referred to as a “regular point”. It is immediately checked that this is what happens in problems of bifurcation
1.3. Introduction to the Mathematical Method......
22
17
from the trivial branch when λ0 is not a characteristic value of L. To
sum up, in the first case, the physical parameter µ can also be used as a
mathematical parameter for the parametrization of the local zero set of
G. The situtaion is different when (3.3) holds instead of (3.2). We can
write
Y = RDµG(0) ⊕ RangeD x G(0),
whereas
e1 = Ker DG(0) = {0} × Ker D xG(0) = {0} × X1 .
X
e1 = 1, we find dim X1 = 1 and the local zero set of G is
Since dim X
a curve parametrized by x1 ǫX1 . It has a “vertical” tangent at the origin,
namely {0} × X1 . Two typical situations are as follows:
''turning point''
''hysteresis point''
Figure 3.2:
Remark 3.1. From a geometric point of view, there is no differences between a “regular point”, a “turning point” or a “hysteresis point”, since
the last two become “regular” after a change of coordinates. But there
is a difference in the number of solutions of the equation G(µ, x) = 0 as
the parameter µ changes sign.
EXAMPLE. Given an element y0 ǫY, let us consider the equation
F(x) = µY 0
23
(3.4)
where F : X → Y is a mapping of class C m (m ≥ 1) such that F(0) = 0.
Setting G(µ, x) = F(x) − µy0 , we are in the first case (i.e. D xG(0) is onto
1. Introduction to One-Parameter Bifurcation Problems
18
) if D x F(0) is onto and the solutions are given by
x = x(µ), x(0) = 0,
where x(·) is a mapping of class C m around the origin. Now, if codim
Range D x F(0) = 1 and y0 < Range D x F(0), we are in the second case.
The curve of solutions has a vertical tangent at the origin and it is not
parametrized by µ; as it follows from the above, a “natural” parameter
e1 = Ker D x F(0) ≥ 1 and
is the one-dimensional variable of the space X
0
y ǫ Range D x F(0), no conclusion can be drawn as yet.
We shall now examine the case n = 1. Here the main tool will be
the Morse lemma, of which we shall give two equivalent formulations.
Theorem 3.1. (Morse lemma : “strong” version)1 : Let f be a mapping
of class C m , m ≥ 2 on a neighbourhood of the origin in R2 with values
in R, such that
f (0) = 0.
D f (0) = 0,
24
det D2 f (0) , 0 (Morse condition).
Then, there is an origin-preserving C m−1 local diffeomorphism φ in R2
with Dφ(0) = I which transforms the local zero set of the quadratic form
e
ξǫR2 → D2 f (0).(e
ξ)2 ǫR,
into the local zero set of f . Moreover, φ is C m away from the origin.
Theorem 3.1’. (Morse lemma, “weak” version): Let f be a mapping of
class C m , m ≥ 2 on a neighbourhood of the origina in R2 with values in
R, verifying
f (0) = 0,
D f (0) = 0,
1
There is an veen stronger version that we shall, however, not use here.
1.3. Introduction to the Mathematical Method......
19
detD2 f (0) , 0(Morse condition).
Then, the local zero set of f reduces to the origin if det D2 f (0) > 0
and is made up of two curves of class C m−1 if det D2 f (0) < 0. Moreover,
these curves are of class C m away from the origin and each of them is
tangent to a different one from among the two lines of the zero set of the
quadratic form
at the origin.
e
ξǫR2 → D2 f (0) · (e
ξ)2 ǫR,
NOTE : We say that the curves intersect transversally at the origin.
COMMENT 3.1. By det D2 f (0), we mean the determinant of the 2 × 2
matrix representing the second derivative of f at the origin for a given
basis of R2 . Of course, this determinant depends on the basis (because
the partial derivatives of f do) but its signs does not (the proof of this
assertion is simple and is left to the reader): the assumptions of Theorem
3.1 and 3.1′ are intrinsically linked to f . In short, we shall say that the
quadratic form D2 f (0) · (e
ξ)2 is non-degenerate.
COMMENT 3.2. Theorem 3.1 implies Theorem 3.1′ , since the local
zero set of the quadratic form
e
ξǫR2 → D2 f (0) · (e
ξ)2 ǫR,
reduces to the origin if det D2 f (0) > 0 (the quadratic form is then positive or negative definite) and is made up of exactly two distinct lines if
det D2 f (0) < 0. If so, the local zero set of f is the image of these two
lines through the diffeomorphism φ: it is then made up of two curves
whose tangents at the origin are the images of the two lines in question
through the linear isomorphism Dφ (0) = I. We shall prove Theorem 3.1′
and that it implies Theorem 3.1′ in the next chapter, in a more general
frame work.
25
1. Introduction to One-Parameter Bifurcation Problems
20
(ii)
(i)
Figure 3.3:
26
COMMENT 3.3. Theorem 3.1 has a generalization to mappings from
Rn+1 → R, n ≥ 1. It is generally stated assuming f ǫC m , m ≥ 3 and the
diffeomorphism φ is shown to be of class C m−2 only (as in Nirenberg
[27]). This improved version is due to Kuiper [21]. Infinite dimensional
versions (cf. [14, 28, 41]) are also available in this direction.
COMMENT 3.4. In contrast, Theorem 3.1′ has a generalization to mapping from Rn+1 → Rn , which we shall prove in the next chapter. This
will be a basic tool in Chapter 3 where we study some general oneparameter problems.
27
Remark 3.2. There is also a generalization of the strong version to mapping from Rn+p → Rn , p ≥ 1 at the expense of losing some regularity at
the origin (cf. [5]).
Application-first results. Assume n = 1 in the Lyapunov-Schmidt
e1 = n +
reduction and let f denote the reduced mapping. Since dim X
1 = 2, dim Y1 = n = 1 and the hypotheses of the Morse lemma are
e1 with R2 , Y1
independent of the system of coordinates, we can identify X
with R so that f becomes a mapping from a neighbourhood of the origin
in R2 with values in R. The conditions ; f (0) = 0, D f (0) = 0 are
automatically fulfilled (cf. (2.10)). If, in addtion, det D2 f (0) , 0 (which
requires G to be of class C 2 at least) the structure of the local zero set of
1.3. Introduction to the Mathematical Method......
21
f , hence that of G follows. In particular, bifurcation occurs as soon as
det D2 f (0) = 0.
Remark 3.3. The independence of the hypotheses of the Morse Lemma
on the system of coordinates does not prove that they are independent
of the spaces X2 and Y1 . However, such a result is true as we shall see
later, in this section.
New formulation of the Morse lemma: We shall prove that the Morse
condition has an equivalent formulation which will be basic for further
generalizations.
Lemma 3.1. Let f be as in the Morse lemma. Then the Morse condition
det D2 f (0) , 0 is equivalent to the following property: For every e
ξǫR2 −
{0} such that D2 f (0).(e
ξ)2 = 0, the linear form D2 f (0) · ξ̃L (R2 , R) is 28
onto.
Proof. As f is a mapping from R2 into R. its derivative D f is a mapping
from R2 into the space L (R2 , R) ≃ R2 . Hence
D2 f (0)ǫL (R2 , L (R2 , R)) ≃ L (R2 , R2 )
and saying that det D2 f (0) , 0 means that Ker D2 f (0) = {0}, i.e. for
every e
ξǫR − {0}, D2 f (0) · e
ξ , 0. But this last condition is certainly
fulfilled by those elements e
ξ such that D2 f (0) · (e
ξ)2 , 0 (regardless of
2
the condition det D f (0) , 0) and it is then not restrictive to write
det D2 f (0) , 0 ⇔ {D2 f (0) · e
ξ , 0 for very e
ξǫR − {0} such that
2
2
e
D f (0) · (ξ) , 0} .
The result follows from the obvious fact that a linear form is onto if
and only ifi it is not the zero form.
With the above lemma, we get equivalent version of the “weak”
Morse lemma.
Theorem 3.1′′. Let f be a mapping of class, C m ≥ 2 on a neighbourhood
of the origin in R2 with values in R, verifying
f (0) = 0,
22
1. Introduction to One-Parameter Bifurcation Problems
D f (0) = 0,
29
such that the linear form D2 f (0) · e
ξ ∈ L (R2 , R) is onto for every ξ̃ ∈
R2 − {0} with D2 f (0) · (ξ̃)2 = 0. Then the local zero set od the quadratic
form D2 f (0) · (e
ξ)2 consists of exactly 0 or 2 real lines (depending on the
sign of det D2 f (0)) and the local zero step of f is made up of the same
number of C m−1 curves through the origin. Moreover, these curves are of
class C m away from the origin and each of them is tangent to a different
one from among the lines in the zero of the quadractic form D2 f (0) · e
ξ ( 2)
at the origine.
Application - further details. Assume that n = 1 in the LyapunovSchmidt reduation and let f denoted the reduced mapping. From the
exposition of Theorem 3.1′′ , it does not matter if R2 and R are replaced
e1 and Y1 respectively. Since f (0) = 0 and D f (0) = 0, it
by the space X
remains to check whether the Morse condition holds. By definition of f
e1 , that
(cf.(2.9)) we first find, for every e
ξǫ X
D f (e
x) · e
ξ = Q1 DG(e
x+e
ϕ(e
x)) · (e
ξ + De
ϕ(e
x) · e
ξ).
Since e
ϕ(0) = 0 and De
ϕ(0) = 0 (cf. (2.12)), we obtain
D2 f (0) · (e
ξ)2 = Q1 D2G(0) · (e
ξ)2 + Q1 DG(0) · (D2 e
ϕ(0) · (e
ξ)2 ).
But Q1 DG(0) = 0, by the definition of Q1 and hence
D2 f (0).(e
ξ)2 = Q1 D2G(0) · (e
ξ)2 .
a particularly simple expression in terms of the mapping G.
We now prove
30
Theorem 3.2. The validity of the Morse condition for the reduced mape2 and Y1 .
ping f is independent of the choice of the spaces X
Proof. Clearly, the mapping
e
e1 → Q1 D2 G(0) · (e
ξǫ X
ξ)2 ǫY1 ,
1.3. Introduction to the Mathematical Method......
23
e2 and it remains to show that the required
is independent of the space X
property of surjectivity is independent of Y1 as well. Let us then assume
that the property holds and let Ŷ1 be another complement of Y2 . Denoting by Q̂1 and Q̂2 the associated projection operators, we must show
e1 − {0} such
that the linear form Q̂1 D2 G(0) · e
ξ is nonzero for every e
ξǫ X
2
2
e
that Q̂1 D G(0) · (ξ) = 0.
First, note that Q̂1 is an isomorphism of Y1 to Ŷ1 . Indeed, both spaces
have the same dimension and it suffices to prove that the restriction of
Q̂1 to Y1 is one-to-one. If Q̂1 y1 = 0 for some y1 ǫY1 , we must have y1 ǫY2
(since Ker Q̂1 − Y2 ) and hence y1 ǫY1 ∩ Y2 = {0}. Next, observe that
Q̂1 = Q̂1 Q1 .
(3.6)
Indeed, one has Q̂1 Q1 = Q̂1 (I − Q2 ) = Q̂1 − Q̂1 Q2 . But Q̂1 Q2 = 0
(since Ker Q̂1 = Y2 again), which proves (3.6).
e1 − {0} be such that Q̂1 D2G(0) · (e
Let then e
ξǫ X
ξ)2 = 0.
From (3.6), this can be written as
Q̂1 Q1 D2G(0) · (e
ξ)2 = 0.
As Q̂1 ǫ Isom (Y1 , Ŷ1 ), we find
Q1 D2G(0) · (e
ξ)2 = 0.
31
But, from our assumptions, Q1 D2G(0)·e
ξ , 0. By the same argument
of isomorphism, Q̂1 Q1 D2G(0) · e
ξ , 0 and using (3.6) again we finally
see that
Q̂1 D2G(0) · e
ξ , 0,
which completes the proof.
Practical Method: Let y0 be any nonzero element of the space Y1− and
let y∗ ǫY ′ (topological dual of Y) be characterized by



 hy∗ , yi = 1,
(3.7)


 hy∗ , yi = 0 for every yǫY2
1. Introduction to One-Parameter Bifurcation Problems
24
(The existence of such an element y∗ is ensured by the Hahn-Banach
theorem). Then, for every yǫY.
Q1 y = hy∗ , yiy0 .
(3.8)
Remark 3.4. One may object that using the Hahn-Banach theorem is
“practical”. Actually, using the linear form y∗ is only a convenient way
to get explicit formulation of the projection operator Q1 , which is the
important thing to know in practice.
e1 , that
It follows, for every e
ξǫ X
32
D2 f (0) · (e
ξ)2 = Q1 D2G(0) · (e
ξ)2 = hy∗ , D2G(0) · (e
ξ)2 iy0 .
e1 so that e
Now let (e
e1 ,e
e2 ) be a basis of X
ξǫe
ξ1 writes
Then
e
ξ = ξ1 e1 + ξ2 e2 , ξ1 , ξ2 ǫR.
Q1 D2 G(0) · (e
ξ)2 = [ξ12 hy∗ , D2G(0) · (e
e1 )2 i + 2ξ1 ξ2 hy∗ , D2G(0) · (e
e1 ,e
e2 )i
+ ξ22 hy∗ , D2 G(0) · (e
e2 )2 i]y0
and the above mapping veriies the Morse condition if and only if the
quadratic form
(ξ1 , ξ2 )ǫR2 →[ξ12 hy∗ , D2G(0) · (e
e1 )2 i + 2ξ1 ξ2 hy∗ , D2G(0) · (e
e1 ,e
e2 )i
+ ξ22 hy∗ , D2G(0) · (e
e2 )2 i]ǫR,
(3.9)
is non-degenerate, i.e. the discriminant
hy∗ , D2 G(0) · (e
e1 ,e
e2 )i2 − hy∗ , D2G(0) · (e
e1 )2 ihy∗ , D2G(0) · (e
e2 )2 i (3.10)
is non zero.
Remark 3.5. Note that the discriminant (3.10) is just (−1) times of the
determinant of the quadratic form (3.9).
1.3. Introduction to the Mathematical Method......
33
25
The example of problems of bifurcation from the trivial branch at a
geometrically simple characteristic value.
Let G(µ, x) = 0 be a problem of bifurcation from the trivial branch
(cf. (1.6)) with compact operator LǫL (X) and nonlinear part ΓǫC m with
m ≥ 2, the real number λ0 being a characteristic value of L (the obvious
case when λ0 is not a characteristic value of L has already been considered). As we know
e1 = R × Ker(I − λ0 L) ⊂ R × X,
X
Y2 = Range(I − λ0 L) ⊂ X(= Y),
so that n = codimY2 = 1 if and only if dim Ker(I − λ0 L) = 1 i.e. the
characteristic value λ0 is geometrically simple.
Since
G(µ, x) = (I − (λ0 + µ)L)x + Γ(µ, x),
we find, from (1.7) and (1.9), that
D2µG(0) = 0
Dµ D xG(0) = −L + Dµ D x Γ(0) = −L
and
D2xG(0) = D2x Γ(0).
As a result, for (µ, x)ǫR × X
D2 G(0) · (µ, x)2 = −2µLx + D2x Γ(0) · (x)2 .
In particular, for xǫX1 , one has Lx = (1/λ0 )x, so that
Q1 D2G(0) · (µ, x)2 = −
2
µQ1 x + Q1 D2xG(0) · (x)2 ,
λ0
for (µ, x)ǫR × Ker(I − λ0 L).
Given any x0 ǫ Ker(I − λ0 L) − {0}, the pair ((1, 0), (0, x0 )) is a basis
of R × Ker(I − λ0 L). The practical method described before leads to the 34
examination of the sign of the determinant of the quadratic polynomial
(µ, t)ǫR2 →
−2µt ∗ 0
hy , x i + t2 hy∗ , D2x Γ(0) · (x0 )2 i,
λ0
(3.11)
26
1. Introduction to One-Parameter Bifurcation Problems
where y∗ ǫX ′ is some linear continuous form with null space Y2 . The
discriminant of the polynomial (3.11) is
4 ∗ 0 2
hy , x i ≥ 0.
λ20
It is positive if and only if hy∗ , x0 i , 0, namely x0 < Y2 . As Ker(I −
λ0 L) = Rx0 and Y2 = Range(I − λ0 L), this means that Ker(I − λ0 L) 1
Range(I − λ0 L). But, if so, (as codim Range (I − λ0 L) = dim Ker(I −
λ0 L) = 1, by hypothesis) we deduce that
X = Ker(I − λ0 L) ⊕ Range(I − λ0 L)
35
(3.12)
which expresses that the characterstic value λ0 is also algebraically simple.
To sum up, the Morse lemma applies to problems of bifurcation
from the trivial branch at a geometrically simple eigenvalue λ0 if and
only if λ0 is also algebraically simple. Then, the local zero set of the
reduced mapping and hence that of G consists of the union of the trivial
branch and a second branch (curve of class C m−1 at the origin and of
class C m away from it) bifurcating from the trivial branch at the origin.
``transcritical'' bifurcation
``supercritical'' bifurcation
Figure 3.4: Local zero set of G.
Remark 3.6. These conclusions agree with Krasnoselskii’s Theorem
(Theorem 1.2) but provide much more precise information on the structure of the local zero set. This result was originally proved by Crandall and Rabinowitz ([7]) in a different way involving the application of
1.3. Introduction to the Mathematical Method......
27
the Implict function theorem, after a modification of the reduced equation. This method uses the fact that the trivial branch is in the local
zero set of G explicitly. Note that the same result holds (with the same
proof) when the more general conditions Γ(µ, 0) = 0, D x Γ(0) = 0 and
Q1 Dµ D x Γ(0) = 0 replace (1.7)-(1.9).
Remark 3.7. The complementary case Ker(I − λ0 L) ⊂ Range(I − λ0 L),
namely, when the characteristic value λ0 is geometrically simple in
which the Morse condition fails (referred to as a “degenerate case”)
will be considered in Chapter 5.
Chapter 2
A generalization of the
Morse Lemma
As We Saw in Chapter 1, the problem is to find the local zero set of 36
the reduced mapping, which is a mapping of class C m (m ≥ 1) in a
neighbourhood of the origin in the (n + 1)-dimensional dspace X1 =
Ker DG(0) into the n-dimensional space Y1 (a given complement of
Y2 = RangeDG(0)).
We shall develop an approach which is analogous to the one we
used in the case n = 1 (Morse lemma). The first task is to find a suitable
generalization of the Morse condition.
2.1 A Nondegeneracy Condition For Homogeneous
Polynomial Mappings.
Let q : Rn+1 → Rn be a polynomial mapping, homogeneous of degree
k ≥ 1 (i.e. q = (qα )α=1,n where qα is a polynomial, homogeneous of
degree k in (n + 1) variables with real coefficients).
Definition 1.1. We shall say that the polynomial mapping q verifies the
condition of R-nondegeneracy (in short, R-N.D.) if, for every non-zero
solution e
ξǫRn+1 of the equation q(e
ξ) = 0, the mapping Dq(ξ)ǫL (Rn+1 ,
n
R ) is onto.
29
30
37
2. A generalization of the Morse Lemma
As is homogeneous, its zero set in Rn+1 is a cone in Rn+1 with vertex
at the origin. Actually, we have much more precise information. First,
observe that
1
ξ) · e
ξ,
(1.1)
q(e
ξ) = Dq(e
k
for every e
ξǫRn+1 (Euler’s theorem). Indeed, from the homogeneity of q,
write
q(te
ξ) = tk q(e
ξ)
and differentiate both sides with respect to t; then
Dq(te
ξ) · e
ξ = ktk−1 q(e
ξ).
Setting t = 1, we get the identity (1.1).
Let e
ξǫRn+1 − {0} be such that q(e
ξ) = 0. Then, it follows that
Re
ξ ⊂ Ker Dq(e
ξ).
But Dq(e
ξ)ǫL (Rn+1 , Rn ) is onto by hypothesis. Hence, dim Ker Dq
e
(ξ) = 1, so that
Ker Dq(e
ξ) = Re
ξ.
(1.2)
Theorem 1.1. Let the polynomial mapping q verify the condition (R −
N.D.). Then, the zero set of q in Rn+1 is made up of a finite number v of
lines through the origin.
Proof. Since the zero set of q is a cone with vertex at the origin, its zero
set is a union of lines. To show that there is a finite number of them, it is
equivalent to showing that their intersection with the unit sphere S n in
Rn+1 consists of a finite number of points. Clearly, the set
38
{e
ξǫS n ; q(e
ξ) = 0},
is closed in S n (continuity of q) and hence compact. To prove that it
is finite, it suffices to show that it is also discrete. Let then e
ξǫS n such
n+1
n
e
e
that q(ξ) = 0. By hypothesis, Dq(ξ)ǫL (R , R ) is onto and we know
that Ker Dq(e
ξ) = Re
ξ (cf. (1.2)). Then, the restriction of Dq(e
ξ) to any
complement of the space Re
ξ is an isomorphism to Rn . In particular,
2.1. A Nondegeneracy Condition For.........
31
observe that the points of the sphere S n have the following property: for
every e
ζǫS n , the tangent space Teζ S n of S n at e
ζ is nothing but {e
ζ}⊥ . hence,
for every e
ζǫS n , we can write
In particular, we deduce
Rn+1 = Re
ζ + Teζ S n .
Dq(e
ξ)ǫ Isom (Teξ S n , Rn ).
As q is regular, the Inverse function theorem shows that there is no
solution other than e
ξ for the equation q(e
ξ) = 0 near e
ξ on S n .
COMMENT 1.1. The above theorem does not prove that there is any
line in the zero set of q. Actually, the situation when ν = 0 can perfectly
occur.
COMMENT 1.2. It is tempting to try to get more information about the
number ν of lines in the zero set of q. Of course, it is not possible to
expect a formula expressing ν in terms of q but one can expect an upper
bound for ν. It can be shown (under the condition (R. N.D.)) that the
inequality
ν ≤ kn ,
(1.3)
always holds. This estimate is an easy application of the generalized 39
Bezout’s theorem (see e. g. Mumford [26]). Its statement will not be
given here because it requires preliminary notions of algebraic geometry
that are beyond the scope of these lectures.
We shall give a flavour of the result by examing the simplest case
n = 1. Let q be a homogeneous polynomial of degree k in two variables.
More precisely, given a basis (e
e1 ,e
e2 ) of R2 , write
then
e
ξ = ξ1e
e1 + ξ2e
e2 , ξ1 , ξ2 ǫR.

k

P



ξ) =
as ξ1k−s ξ2s ,
 q(e
s=0




 as ǫR, 0 ≤ s ≤ k.
(1.4)
2. A generalization of the Morse Lemma
32
It is well-known that such a polynomial q can be deduced from a
unique k-linear symmetric form Q on R2 by
q(e
ξ) = Q(e
ξ, · · · , e
ξ),
where the argument e
ξǫR2 is repeated k times. In particular,
!
k
as =
Q(e
e1 , · · · ,e
e1 ,e
e2 , · · · ,e
e2 ),
s
where the argument e
e1 (respectively e
e2 ) is repeated s times (respectively
k − 1 times). Now, the basis (e
e1 ,e
e2 ) can be chosen so that ak , 0. Indeed
ak = q(e
e2 , · · · ,e
e2 ) = q(e
e2 )
40
and e
e2 can be taken so that q(e
e2 ) , 0 (since q . 0),e
e1 being any vector
in R2 , not collinear with e
e2 . If so, observe that the local zero set of
q contains no element of the form ξ2e
e2 , ξ2 , 0, since q(ξ2e
e2 ) = ak ξ2k .
In other words, each nonzero solution of the equation q(e
ξ) = 0 has a
k
nonzero component ξ1 . Dividing then (1.4) by ξ1 , we find
q(e
ξ) = 0 ⇔
k
X
as
s=0
ξ2
ξ1
!s
= 0.
Setting τ = ξξ12 and since τ is real whenever ξ1 and ξ2 are, we see that
τ must be a real root of the polynomial
a(τ) =
k
X
as τs .
s=0
Conversely, to each real root τ of the above polynomial is associated
the line {ξ1e
e1 + τξ1e
e2 ; ξ1 ǫR} of solutions of the equation q(e
ξ) = 0. Here,
the inequality ν ≤ k follwos from the fundamental theorem of algebra.
Remark 1.1. Writing
a(τ) =
k
X
s=0
as τ s = ak
k
Y
s=1
(τ − τ s )
2.2. Practical Verification of the Condition (R-N.D.).
33
where τ s , 1 ≤ s ≤ k are the k (not necessarily distinct) roots of the
polynomial a(τ) and replacing τ by ξ2 |ξ1 with ξ1 , 0, we find
q(e
ξ) = ak
k
Y
(ξ2 − τ s ξ1 ),
s=1
as relation which remains valid when ξ1 = 0.
COMMENT 1.3. Recall that a continuous odd mapping defined on the 41
sphere S m−1 ⊂ Rm with values in Rn always vanishes at some point of
S m−1 when m > n. Here, with m = n + 1 we deduce that ν ≥ 1 when
k is odd. When k is even, it can be shown (cf. Buchner, Marsden and
Schecter [5]) taht ν is even too (possibly 0 however).
Remark 1.2. Any small perturbation of q (as a homogeneous polynomial mapping of degree k) still verifies the condition (R - N.D.) and its
local zero set id made of the same number of lines (Hint: let Q denote
the finite dimensional space of homogeneous polynomials of degree k.
Consider the mapping (p, e
ζ)ǫQ×S n → p(e
ζ)ǫRn and note that the derivative at (q, e
ξ) with respect to e
ζ is an isomorphism when q(e
ξ) = 0.)
COMMENT 1.4. Condition (R-N.D.) ensures that the zero set of q in
Rn+1 is made of a finite number of lines through the origin. The converse
is not true1 . Actually, the condition (R-N.D.) also shows that each line
in the zero set is “simple” in the way described in Remark 1.2. When
the condition (R.N.D.) does not hold but the zero set of q is still made up
of a finite number of lines, some of them are “multiple”, namely, split
into several lines or else disappear when replacing q by a suitable small
perturbation.
2.2 Practical Verification of the Condition (R-N.D.).
The above considerations leave us with two basic questions:
(i) How does one check the condition (R-N.D.) for a given mapping 42
1
Incidentlly, we have shown that the zero set of q is a finite union of lines, when
n = 1, with no assumption other than q . 0.
2. A generalization of the Morse Lemma
34
q?
(ii) If the condition (R-N.D.) holds, how can we compute (approximations of) the lines in the zero set of q?
Here we shall give a partial answer to these questions. We begin
with the case n = 1. As we know, in every system of coordinates e
ξ =
2
ξ1e
e1 + ξ2e
e2 of R such that q(e
e2 ) , 0 (such a system always exists when
q . 0), one has
k
Y
q(e
ξ) = ak
(ξ2 − τ s ξ1 ),
(2.1)
s=1
where ak = q(e
e2 ) , 0 and τ s , 1 ≤ s ≤ k are the k roots of the polynomial
a(τ) =
k
X
as τs .
(2.2)
s=0
If so, each line in the zero set of q is of the form
{ξ1e
e1 + τ s ξ1e
e1 ; ξ1 ǫR},
(2.3)
where τ s is a real root of a(τ). Saying that q verifies the condition (RN.D.) amounts to saying that the derivative Dq(e
ξ)ǫL (R2 , R) is onto (i.e.
not equal to zero) at each point e
ξǫR2 −{0} such that q(e
ξ) = 0. For e
ξ, e
ζǫR2 ,

k Y
X



(2.4)
Dq(e
ξ) · e
ζ = ak
 (ξ2 − τσ ξ1 ) (ζ2 − τ s ζ1 ).
s=1 σ,s
Since q(e
ξ) = 0, there is an index s0 such that ξ2 − τ s0 ξ1 = 0. Thus



 Y
(τ s0 ξ1 − τσ ξ1 ) (ζ2 − τ s0 ζ1 ).
Dq(e
ξ) · e
ζ = ak 
σ,s0
43
Clearly, the linear form e
ζǫR2 → ζ2 − τ s0 ζ1 is onto (i.e. non-zero).
Thus, Dq(e
ξ) will be onto if and only if
Y
ak
(2.5)
(τ s0 − τσ )ξ1 , 0.
σ,s0
2.2. Practical Verification of the Condition (R-N.D.).
35
But from the choice of the system of coordinates, we know that
ξ1 , 0. Hence, Dq(e
ξ) will be onto if and only if
τσ , τ s0 for σ , s0 ,
i.e. τ s0 is a simple root of the polynomial a(τ).
To sum up, when n = 1, the mapping q will verify the condition
(R-N.D.) if and only if given a system of coordinates (e
e1 ,e
e2 ) in R2 such
k
P
that q(e
e2 )(= ak ) , 0, each real root of the polynomial a(τ) =
as τ s is
s=0
simple.
In the particular case when k = 2, a(τ) is a quadratic polynomial.
(i) If its discriminant is < 0, it has no real root (then, each of them is
simple) : (R-N.D.) holds.
(ii) If its discriminant is zero, it has a double real root: (R-N.D.) fails
to hold.
(iii) If its discriminant is > 0, it has two simple real roots: (R-N.D.)
holds.
Note that the above results give a way for finding (approximations
of) the lines in the zero set of q. It suffices to use an algorithm for the 44
computation of the roots of the polynomial a(τ). However, it is not easy
to check whether a given root of a polynomial is simple, by calculating
approximations to it through an algorithm and the first question is not
satisfactorily answered.
Observe that it is of course sufficient, for the condition (R-N.D.) to
hold , that every root (real or complex) of the polynomial a(τ) is simple.
Definition 2.1. If every root (real or comples) of a(τ) is simple, we shall
say that the polynomial q satisfies the condition of C-nondegeneracy (in
short, C-N.D.)
Definition 2.1 can be described by saying that the polynomial a(τ)
and its derivative a′ (τ) have no common root.
2. A generalization of the Morse Lemma
36
Now recall the following result from elementary algebra: let a(τ)
and b(τ) be two polynomials (with complex coefficients) of degrees exactly k and ℓ respectively, i.e.

k

P



a(τ) =
as , τ s , ak , 0,



s=0
(2.6)


ℓ
P


s

bs τ , bℓ , 0.

 b(τ) =
s=0
45
Then, a necessary and sufficient condition for a(τ) and b(τ) to have
a common root in C is that the (k + ℓ) × (k + ℓ) determinant
ak · · · a0
ak
· · · a0
···
ak
· · · a0 R = bℓ · · · b0
bℓ
· · · b0
···
bℓ
· · · b0 in which there are ℓ rows of “a” entries and k rows of “b” entries, is
0 (note that this statement is false in R). The determinant R is called the
resultant (of Sylvester) of a(τ) and b(τ). In particular, when b(τ) = a′ (τ)
(so that ℓ = k − 1), R is called the discriminant, denoter by D, of a(τ).
It is a (2k − 1) × (2k − 1) determinant. Elementary properties and further
developments about resultants can be found in Hodge and Pedoe [16] or
Kendig [18].
Again, we examine the simple case when k = 2. If so,
a(τ) = a2 τ2 + a1 τ + a0 , a2 , 0,
so that
a′ (τ) = 2a2 τ + a1 .
Now, from the definitions
a2 a1
D = 2a2 a1
0 2a
2
a0 0 = −a2 (a21 − 4a0 a2 ).
a1 2.2. Practical Verification of the Condition (R-N.D.).
37
The quantity a21 − 4a0 a2 is the usual discriminant and, as a2 , 0 we 46
conclude
D , 0 ⇔ a21 − 4a2 a0 , 0
(2.8)
Remark 2.1. When k = 2, the condition (R - N.D.) is characterized by
saying that the discriminant is , 0 too. Hence, when k = 2
(R − N.D.) ⇔ (C − N.D.),
(2.9)
but this is no longer true for k ≧ 4.
We have found that the condition (C − N.D.) holds ⇔ D , 0. The
advantage of this stronger assumption is that it is immediate to obtain
the discriminant D in terms of the coefficients as ’s, hence from q.
Expression of the conditions (R− N.D.) and (C− N.D.) in any system
of coordinates: We know to express the coorditions (R − N.D.) and (C −
N.D.) in a system of coordinates e
e1 ,e
e2 such that q(e
e2 ) , 0. Actually,
we can get such an expression in any system of coordinates. Indeed,
assume q(e
e2 ) = 0. Then, the coefficient ak vanishes and we have
q(e
ξ) =
k−1
X
as ξ1k−1 ξ2s .
(2.10)
s=0
It follows that the line {ξ2e
e2 , ξ2 ǫR} is in the zero set of q. Away from 47
the origin on this line, the derivative Dq(e
ξ) must be onto (i.e. , 0). An
immediate calculation shows, for e
ξ = ξ2e
e2 , that
Dq(e
ξ) · e
ζ = ak−1 ξ2k−1 ζ1 ,
(2.11)
for every e
ζǫR2 . As e
ξ is , 0 if and only if ξ2 is , 0, we have
Dq(e
ξ) is onto ⇔ ak−1 , 0.
Now, for any solution e
ξ of the equation q(e
ξ) = 0 which is not on the
line Rẽ2 , we must have ξ1 , 0. Arguing as before, we get
q(e
ξ) =
ξ1k
k−1
X
s=0
as
ξ2
ξ1
!s
2. A generalization of the Morse Lemma
38
and all this amounts to solving the equation,
k−1
X
as
s=0
ξ2
ξ1
!2
= 0.
Setting τ = ξ2 /ξ1 and
a(τ) =
k−1
X
as τs
s=0
the method we used when ak , 0 shows that the condition (R − N.D.) is
equivalent to the fact that each real root of a(τ) is simple. The condition
(C − N.D.) being expressed by saying that every root of a(τ) is simple,
can be written as
D , 0,
where D is the discriminant of a(τ).
48
Remark 2.2. As ak = 0, the polynomial a(τ) must be considered as a
polynoimal of degree k − 1, namely D is a (2k − 3) × (2k − 3) determinant
(instead of (2k − 1) × (2k − 1) when a(τ) is of degree k). If a(τ) is
considered as a polynomial of degree k with leading coefficient equal to
zero, the determinant we obtain is always zero and has no significance.
Now, we come back to the general case when n is arbitrary. Motivated by the results when n = 1, it is interesting to look for a generalization of the condition (C − N.D.). Let Q be the k-linear symmetric
mapping such that
q(e
ξ) = Q(e
ξ, · · · , e
ξ),
where the argument e
ξ is repeated k times. Then Q has a canonical extension as a k-linear symmetric mapping from Cn+1 → Cn (so that linear
means C-linear here). Indeed, Cn+1 and Cn identify with Rn+1 + iRn+1
and Rn + iRn respectively. Now, take k elements e
ξ (1) , · · · , e
ξ (k) in Cn+1 .
These elements can be written as
e
ξ (s) = e
u(s) + ie
v(s) , e
u(s) ,e
v(s) ǫRn+1 , 1 ≤ s ≤ k.
2.2. Practical Verification of the Condition (R-N.D.).
39
Define the extension of Q (still denoted by Q) by
Q(e
ξ (1) , · · · , e
ξ (k) ) =
k
X
j=1
ik− j
X
PǫP j
Q(e
u(P(1)) , e
u(P( j)) ,e
v(P( j+1)) ,e
v(P(k)) ),
where P j denotes the set of permutations of {1, · · · , k} such that
P(1) < · < P( j)
and
P( j + 1) < · < P(k).
It is easily checked that this defines an extension of Q which is C- 49
linear with respect to each argument and symmetric. An extension of q
to Cn+1 is then
q(e
ξ) = Q(e
ξ, · · · , e
ξ).
(2.12)
Remark 2.3. In practice, if q = (qα )α=1,··· ,n and
e
ξ = ξ1e
e1 + · · · + ξn+1e
en+1 ,
where (e
e1 , · · · ,e
en+1 ) is a basis of Rn+1 with ξ1 , · · · , ξn+1 ǫR, each qα is a
polynomial, homogeneous of degree k with real coefficients. The extension (2.12) is obtained by simply replacing each ξ j ǫR by ξ j ǫC.
Definition 2.2. We shall say that q verifies the condition of C-non - degenreacy (in short, (C − N.D.)) if, for every non-zero solution e
ξ of the
equation q(e
ξ) = 0, the linear mapping Dq(e
ξ)ǫL (Cn+1 , Cn ) (complex
derivative) is onto.
Of course, when n = 1, Definition 2.2 coincided with Definition 2.1.
Whenever the mapping q verifies the condition (C − N.D.), it verifies the
condition (R − N.D.) as well: This is immediately checked after observing for e
ξǫRn+1 that the restriction to Rn+1 of the complex derivative of q
at e
ξ is nothing but its real derivative. A method for checking the conditions (R − N.D.) and (C − N.D.) when n = 2 is described in Appendix
1 where we also make some comments on the general case, not quite
completely solved however.
40
2. A generalization of the Morse Lemma
2.3 A Generalization of the Morse Lemma for Mappings from Rn+1 into Rn : “Weak” Regularity
Results.
50
From now on, the space Rn+1 is equipped with its euclidean structure.
Let O be an open neighbourhood of the origin in Rn+1 and
f : O → Rn ,
a mapping of class C m , m ≥ 1. Assume there is a positive integer k, 1 ≤
k ≤ m such that
D j f (0) = 0 0 ≤ j ≤ k − 1
(3.1)
(in particular f (0) = 0). For every e
ξǫRn+1 , set
q(e
ξ) = Dk f (0) · (e
ξ)k .
(3.2)
Our purpose is to give a precise description of the zero set of f
around the origin (local zero set). We may limit ourselves to seeking
nonzero solution only. For this, we first perform a transformation of
the problem. Let r0 > 0 be such that the closed ball B(0, r0 ) in Rn+1 is
contained in O. The problem will be solved if, for some r, 0 ≤ r ≤ r0
and every 0 < |t| < r, we are able to determine the solutions of the
equation
f (e
x) = 0, ||e
x|| = |t|.
51
It is immediate that e
ξ is a solution for this system if and only if we
can write
e
x = te
ξ
with



xǫS n ,
 0 < |t| < r, e


 f (te
x) = 0
where S n is the unit sphere in Rn+1 . Also it is not restrictive to assume
that f is defined in the whole space Rn+1 (Indeed, f can always be extended as a C m mapping outside B(0, r0 )).
2.3. A Generalization of the Morse Lemma.....
41
Now, let us define
g : R × Rn+1 → Rn ,
by
g(t, e
ξ) = k
Z
1
0
(1 − s)k−1 Dk f (ste
ξ) · (e
ξ)k ds,
(3.3)
a fromula showing that g is of class C m−k in R × Rn+1 .
Lemma 3.1. Let g be defined as above. Then
k! e
f (tξ) for t , 0, e
ξǫRn+1 ,
tk
g(0, e
ξ) = Dk f (0) · (e
ξ)k for e
ξǫRn+1 .
g(t, e
ξ) =
(3.4)
(3.5)
Proof. The relation g(0, ·) = q follows from the definitions immediately.
Next for e
xǫRn+1 , write the Taylor expansion of order k − 1 of f about
the origin. Due to (3.1),
1
f (e
x) =
(k − 1)!
Z
0
1
(1 − s)k−1 Dk f (se
x) · (e
x)k ds.
With e
x = te
ξ, t , 0 and comparing with (3.3), we find
g(t, e
ξ) =
k! e
f (tξ).
tk
From Lemma 3.1 the problem is equivalent to



ξǫS n ,
 0 < |t| < r, e


 g(t, e
ξ) = 0.
In what follows, we shall solve the equation (for small enough r > 0)



ξǫS n ,
 tǫ(−r, r), e
(3.6)


 g(t, e
ξ) = 0.
52
2. A generalization of the Morse Lemma
42
Its solutions (t, e
ξ) with t , 0 will provide the nonzero solutions
of f (e
x) = 0 verifying ||e
x|| = |t| through the simple relation e
x = te
ξ.
Of course , the trivial solution e
x = 0 is also obtained as e
x = 0e
ξ with
g(0, e
ξ) = q(e
ξ) = 0 unless this equation has no solution on the unit sphere.
Thus all the solutions of f (e
x) = 0 such that 0 < ||e
x|| < r are given by
e
e
e
x = tξ with (t, ξ) solution of (3.6) provided that the zero set of q does
not reduce to the origin.
From now on, we assume that the mapping q verifies the condition
(R − N.D.) and we denote by ν ≥ 0 the number of lines in the zero set of
q, so that the set
{e
ξǫS n ; q(e
ξ) = 0},
(3.7)
has exactly 2ν elements. We shall set
53
{e
ξǫS n ; q(e
ξ) = 0} = {e
ξ01 , · · · , e
ξ02ν },
with an obvious abuse of notation when ν = 0. This set is stable under
j
multiplication by −1, so that we may assume that the e
ξ0 ’s have been
arranged so that
e
ξ0ν+ j = −e
ξ0j , 1 ≤ j ≤ ν.
(3.8)
Lemma 3.2. (i) Assume ν ≥ 1 and for each index 1 ≤ j ≤ 2ν, let
σ j ⊂ S n denote a neighbourhood of e
ξ0j . Then, there exists 0 < r ≤ r0
such that the conditions (t, e
ξ)ǫ(−r, r) × S n and g(t, e
ξ) = 0 together imply
e
ξǫ
2ν
[
σ j.
j=1
(ii) Assume that ν = 0. Then, there exists 0 < r < r0 such that the
equation g(t, e
ξ) = 0 has no solution in the set (−r, r) × S n .
Proof. (i) We argue by contradiction : If not, there is a sequence (tℓ ,
e
ξℓ )ℓ≥1 with lim tℓ = 0 and e
ξℓ ǫS n such that
ℓ→∞
and
g(tℓ , e
ξℓ ) = 0
e
ξℓ <
2ν
[
j=1
σ j.
2.3. A Generalization of the Morse Lemma.....
43
From the compactness of S n and after considering a subsequence,
we may assume that there exists e
ξǫS n such that lim e
ξℓ = e
ξ. By the
ℓ→+∞
continuity of g, g(0, e
ξ) = 0. As g(0, ·) = q, e
ξ must be one of the elements
j
e
ξ0 , which is impossible since
e
ξℓ <
2ν
[
σ j,
j=1
for every ℓ ≥ 1 so that the sequence (e
ξℓ ) cannot converge to e
ξ.
54
(ii) Again we argue by contradiction. If there is a sequence (tℓ , e
ξℓ )ℓ≥1
such that lim tℓ = 0, e
ξℓ ǫS n and g(tℓ , e
ξℓ ) = 0, the continuity of g and the
ℓ→+∞
compactness of S n show that there is e
ξǫS n verifying q(e
ξ) = g(0, e
ξ) = 0
and we reach a contradiction with the hypothesis ν = 0.
Remark 3.1. From Lemma 3.2, the equation f (e
x) = 0 has then no solution e
x , 0 in a sufficiently small neighbourhood of the origin when
ν = 0; in other words, the local zero set of f reduces to the origin.
We shall then focus on the main case when ν ≥ 1. For this we need
the following lemma.
Lemma 3.3. The mapping g verifies
gǫC m−k (R × Rn+1 , Rn )
ξ) exists for every pair (t, e
ξ)ǫR × Rn+1 .
and the partial derivative Deξ g(t, e
Moreover
Deξ gǫC m−k (R × Rn+1 , L (Rn+1 , Rn ))
Proof. We already know that gǫC m−k . Besides, the existence of a partial
derivative Deξ g(t, e
ξ) at every point (t, e
ξ)ǫR × Rn+1 is obvious from the
relations (3.4) and (3.5), from which we get
and
Deξ g(t, e
ξ) =
k!
D f (te
ξ) if t , 0,
tk−1
Deξ g(0, e
ξ) = Dq(e
ξ) = kDk f (0) · (e
ξ)k−1 .
(3.9)
(3.10)
2. A generalization of the Morse Lemma
44
55
First, assume that k = 1. Then
Deξ g(t, e
ξ) = D f (t, e
ξ),
Deξ g(0, e
ξ) = D f (0)
and the assertion follows from the fact that D f is C m−1 , by hypothesis.
Now, assume k ≥ 2 and write the Taylor expansion of D f of order k − 2
about the origin. For every e
xǫRn+1 and due to (3.1)
1
D f (e
x) =
(k − 2)!
With e
x = te
ξ,
D f (te
ξ) =
tk−1
(k − 2)!
Z
1
(1 − s)k−2 Dk f (se
x) · (e
x)k−1 ds.
0
Z
1
0
(1 − s)k−2 Dk f (ste
ξ) · (e
ξ)k−1 ds.
From (3.9) and (3.10), the relation
ξ) = k(k − 1)
Deξ g(t, e
Z
1
0
(1 − s)k−2 Dk f (ste
ξ) · (e
ξ)k−1 ds
holds for every tǫR and every e
ξǫRn+1 . Hence the result, since the right
hand side of this identity is of class C m−k .
Finally, let us recall the so-called “strong” version of the Implicit
function theorem (see Lyusternik and Sobolev [22]).
56
Lemma 3.4. Let U, V and W be real Banach spaces and F = (F(u, v))
a mapping defined on a neighbourhood O of the origin in U × V with
values in W. Assume F(0) = 0 and
(i) F is continuous in O,
(ii) the derivative Dv F is defined and continuous in O,
(iii) Dv F(0)ǫI som(V, W).
2.3. A Generalization of the Morse Lemma.....
45
Then, the zero set of F around the origin in U × V coincides with the
graph of a continuous function defined in a neighbourhood of the origin
in U with values in V.
Remark 3.2. In the above statement, F is not supposed to be C 1 and
the result is weaker than in the usual Implict function theorem. The
function whoce graph is the zero set of F around the origin is found to
be poly continuous (instead of C 1 ). The proof of this “strong” version
is the same as the proof of the usual statement after observing that the
assumptions of Lemma 3.4 are sufficient to prove continuity.
We can now state an important result on the structure of the solutions
of the equation (3.6).
Theorem 3.1. Assume ν ≥ 1; then, there exists r > 0 such that the
equation
g(t, e
ξ) = 0, (t, e
ξ)ǫ(−r, r) × S n
is equivalent to
tǫ(−r, r), e
ξ =e
ξ j (t),
for some index 1 ≤ j ≤ 2ν where, for each index 1 ≤ j ≤ 2ν, the function 57
e
ξ j is of class C m−k from (−r, r) into S n and is uniquely determined. In
particular,
j
e
ξ j (0) = e
ξ0 , 1 ≤ j ≤ 2ν,
and
e
ξ v+ j (t) = −e
ξ j(−t),
for every 1 ≤ j ≤ 2ν and every tǫ(−r, r).
Proof. We first solve the equation g(t, e
ξ) = 0 around the solution (t =
j
e
e
0, ξ = ξ0 ) for each index 1 ≤ j ≤ 2ν separately. Let us then fix 1 ≤ j ≤
2ν. As we in Chapter 2, 2.1, the condition (R − N.D.) allows us to write
j
Rn+1 = Ker Dq(e
ξ0 ) ⊕ Teξ j S n .
0
From (3.5),
j
j
Dq(e
ξ0 ) = Deξ g(0, e
ξ0 )
(3.11)
2. A generalization of the Morse Lemma
46
and (3.11) shows that
j
Deξ g(0, e
ξ0 )|Tej S n ǫI som(Teξ j S n , Rn ).
ξ
(3.12)
0
0
j
′
e
Let then θ−1
j (θ j = θ j (ξ )) be a chart around ξ0 , centered at the origin
j
of Rn (i.e. θ(0) = e
ξ ) and set
0
ĝ(t, ξ ′ ) = g(t, θ j (ξ ′ )).
Then Dξ , ĝ is defined around (t = 0, ξ ′ = 0) and
58
Dξ , ĝ(t, ξ ′ ) = Deξ g(t, θ j (ξ ′ )) · Dθ j (ξ ′ )
From Lemma 3.3, it follows that ĝ and Dξ , ĝ are of class C m−k
around the origin in R × Rn . Besides,
j
j
ξ0 ) = 0
ĝ(0) = g(0, e
ξ0 ) = q(e
and combining (3.12) with the fact that Dθ j (0) is an isomorphism of Rn
to Teξ j S n (recall that θ−1
j is a chart), one has
0
Dξ , ĝ(0)ǫI som(Rn , Rn ).
If m − k ≥ 1, the Implicit function theorem (usual version) states that
the zero set of ĝ around the origin is the graph of a (necessarily unique)
mapping
t → ξ ′ (t)
of class C m−k around the origin verifying ξ ′ (0) = 0. If m − k = 0, the
same result holds by using the “strong” version of the Implict function
theorem (Lemma 3.4). The zero of g around the point (0, e
ξ0j ) is then the
graph of the mapping
t → θ j (ξ ′ (t)),
of class C m−k around the origin, which is the desired mapping e
ξ j (t). In
particular, given any sufficiently small r > 0 and any sufficiently small
j
neighbourhood σ j of e
ξ0 in S n , there are no solutions of the equation
g(t, e
ξ) = 0 in (−r, r) × σ j other than those of the form (t, e
ξ j (t)).
2.3. A Generalization of the Morse Lemma.....
59
47
At this stage, note that the above property is not affected by arbitrarily shrinking r > 0. In particular, the same r > 0 can be chosen
for every index 1 ≤ j ≤ 2ν. Further, applying Lemma 3.2 with disjoint neighbourhoods σ j , 1 ≤ j ≤ 2ν, we see after shrinking r again,
if necessary, that there in no solution of the equation g(t, e
ξ) = 0 with
2ν
S
|t| < r outside
σ j . In other words, we have e
ξ j (t)ǫσ j , 1 ≤ j ≤ 2ν
j=1
and the solutions of g(t, e
ξ) = 0 with |t| < r are exactly the 2ν pairs
(t, e
ξ j (t)), 1 ≤ j ≤ 2ν. These pairs are distinct, since the neighbourhoods
σ j we have been taken disjoint.
Finally, writing g(−t, e
ξ j (−t)) = 0 for |t| < r and observing that
g(−t, −e
ξ) = g(t, e
ξ), we find g(t, −e
ξ j (−t)) = 0 for |t| < r. Since −e
ξ j (0) =
j
ν+
j
−e
ξ0 = e
ξ0 for 1 ≤ j ≤ ν (cf. (3.8)), we deduce that the function −e
ξ j (−t)
has the property characterizing e
ξ ν+ j (t) and the proof is complete.
Corollary 3.1. Under our assumptions, the equation f (e
x) = 0 has no
solutions (e
x) , 0 around the origin in Rn+1 when ν = 0. When ν ≥ 1
and for r > 0 small enough, the solutions of the equation f (e
x) = 0
with ||e
x|| < r are given by e
x = e
x j (t), 1 ≤ j ≤ ν, where the functions
e
x j ǫC m−k ((−r, r), Rn+1 ) are defined through the functions e
ξ, 1 ≤ j ≤ ν of
Theorem 3.1 by the formula
e
x j (t) = te
ξ j (t), tǫ(−r, r).
In addition, the functions e
x j , 1 ≤ j ≤ ν, are differentiable at the
origin with
de
x j ej
= ξ , 1 ≤ j ≤ ν.
dt
Proof. We already observed in Remark 3.1 that the local zero set of f 60
reduces to {0} when ν = 0. Assume then ν ≥ 1. We know that the
solution of f (e
x) = 0 with ||e
x|| < r are of the form e
x = te
ξ with |t| < r,
e
e
ξǫS n , g(t, ξ) = 0. From Theorem 3.1, r > 0 can be chosen so that
e
x=e
x j (t) = te
ξ j (t), 1 ≤ j ≤ 2ν.
Like e
ξ j , each function e
x j is of class C m−k . In addition, for t , 0,
e
x j (t) ej
= ξ (t),
t
2. A generalization of the Morse Lemma
48
so that
de
xj
(0) exists with
dt
de
xj
(0) = lim e
ξ j (t) = e
ξ0j .
t→0
dt
Finally, from the relation e
ξ ν+ j (t) = −e
ξ j (t), for 1 ≤ j ≤ ν, we get
e
xν+ j (t) = e
x j (t), 1 ≤ j ≤ ν,
and the solution of f (x) = 0 are given through the first ν functions e
xj
only.
2.4 Further Regularity Results.
With Corollary 3.1 as a starting point, we shall now show, without any
additional assumption, that the functions e
x j are actually of class C m−k+1
m
at the and of class C away from it. This latter assertion is the simpler
one to prove.
61
Lemma 4.1. After shrinking r > 0 if necessary, the functions e
x j, 1 ≤
m
j ≤ ν are of class C on (−r, r) − {0}.
Proof. From the relation e
x j (t) = te
ξ j (t), it is clear that the functions e
xj
j
and e
ξ have the same regularity away from the origin. Thus, we shall
show that the functions e
ξ j are of class C m away from the origin. Recall
j
e
that ξ is characterized by


e

ξ j (t)ǫS n




g(t, e
ξ j (t)) = 0,




j

 e
ξ j (0) = e
ξ0 ,
for tǫ(−r, r). Also recall (cf. (3.12))
Deξ g(0, e
ξ0j ) = Dq(e
ξ0j )ǫI som(Teξ j S n , Rn ).
0
Then, by the continuity of Deξ g (Lemma 3.3), there is an open neighj
bourhood σ j of e
ξ in S n such that, after shrinking r > 0 if necessary,
0
ξ)ǫI som(Teξ S n , Rn )
Deξ g(t, e
(4.1)
2.4. Further Regularity Results.
49
for every (t, e
ξ)ǫ(−r, r) × σ j . By shrinking r > 0 again and due to the
continuity of the function e
ξ j, we can suppose that e
ξ j takes its values in
σ j for tǫ(−r, r). Let then t0 ǫ(−r, r), t0 , 0. Thus
g(t0 , e
ξ j(t0 )) = 0,
and, from (4.1),
Deξ g(t0 , e
ξ j (t0 ))ǫI som(Teξ j (t0 ) S n , Rn ).
62
As t0 , 0, g(t, e
ξ) is given by (3.4) for t around t0 and e
ξǫRn+1 and thus
the mapping g has the same regularity as f (i.e. is of class C m ) around
the point (t0 , e
ξ j (t0 )). By the Implicit function theorem, we find that the
zero set of g around (t0 , e
ξ(t0 )) in (−r, r) × S n coincides with the graph of
a unique function e
ζ =e
ζ(t) of class C m around t0 , such that e
ζ j (t0 ) = e
ξ(t0 ).
j
e
e
But from the uniqueness, we must have ζ(t) = ξ (t) around t = t0 so that
the function e
ξ j (·) is of class C m around t0 for every t0 , 0 in (−r.r). To prove the regularity C m−k+1 of the function e
x j at the origin, we
shall introduce the mapping
h : R × Rn+1 → Rn ,
defined by
if k = 1 and by
h(t, e
ξ) = D f (te
ξ) · e
ξ−
h(t, ξ) = k
Z
0
1
Z
0
1
D f (ste
ξ) · e
ξds,
d
[−s(1 − s)k−1 ]Dk f (ste
ξ) · (e
ξ)k ds,
ds
(4.2)
(4.3)
if k ≥ 2. Since f is of class C m , it is clear, in any case, that hǫC m−k (R ×
Rn+1 , Rn ). In addition, with t = 0 in the definition of h((4.3)), we find
h(0, e
ξ) = 0 for every e
ξǫRn+1 .
(4.4)
2. A generalization of the Morse Lemma
50
Lemma 4.2. For any (t, e
ξ)ǫ(R−{0})×Rn+1 , the partial derivative (∂g/∂t) 63
e
(t, ξ) exists and
∂q e 1
(t, ξ) = h(t, e
ξ).
(4.5)
∂t
t
∂q e
(t, ξ) for t , 0 immediate from the relation
Proof. The existence of
∂t
(3.4) from which we get
∂q e k!
k
(t, ξ) = k (D f (te
ξ)).
ξ) · e
ξ − f (te
∂t
t
t
Hence
t∂q e
k
k!
ξ) · e
ξ − f (te
(t, ξ) = k−1 (D f (te
ξ)).
∂t
t
t
If k = 1, the relation (4.6) becomes
Writing
(4.6)
t∂q e
1
(t, ξ) = D f (te
ξ) · e
ξ − f (te
ξ).
∂t
t
f (te
ξ) = t
Z
0
1
D f (ste
ξ) · e
ξds,
the desired relation (4.5) follows from (4.2).
Now, assume k ≥ 2. Recall the relations
Z 1
k
t
(1 − s)k−1 Dk f (ste
ξ) · (e
ξ)k ds,
f (te
ξ) =
(k − 1)! 0
Z 1
tk−1
e
D f (tξ) =
(1 − s)k−2 Dk f (ste
ξ) · (e
ξ)k−1 ds,
(k − 2)! 0
64
that we have already used in the proofs of Lemma 3.1 and Lemma 3.3
respectively. After an immediate calculation, (4.6) becomes
Z 1
∂q
ξ) = k
t (t, e
[(k − 1)(1 − s)k−2 ]Dk f (ste
ξ) · (e
ξ)k ds.
∂t
0
But
i
d h
−s(1 − s)k−1 ,
ds
so that (4.5) follows from the definition (4.3) of h.
(k − 1)(1 − s)k−2 − k(1 − s)k−1 =
2.4. Further Regularity Results.
51
Theorem 4.1. (Structure of the local zero set of f ): Assume ν ≥ 1. For
sufficiently small, r > 0 the local zero set of f in the ball B(0, r) ⊂ Rn+1
consists of exactly ν curves of class C m−k+1 at the origin and class C m
away from the origin. These curves are tangent to a different one from
among the ν lines in the zero set of q(e
ξ) = Dk f (0) · (e
ξ)k at the origin.
Proof. In Lemma 4.1, we proved that the functions e
x j and e
ξ j are of class
C m away from the origin. Since m ≥ 1, we may differentiate the identity
e
x j (t) = te
ξ j(t),
to get
de
ξ j (t) ej
de
x j (t)
=t
+ ξ (t), 0 < |t| < r.
dt
dt
Also, we know that e
x j is differentiable at the origin with
(4.7)
de
xj
(0) = e
ξ0j .
dt
We shall prove that the function (de
x j /dt) is of class C m−k at the
origin. Let σ j ⊂ S n be the open neighbourhood of e
ξ0j considered in
Lemma 4.1, so that Deξ g(t, e
ξ)ǫI som(Teξ S n , Rn ) for every (t, e
ξ)ǫ(−r, r)×σ j . 65
h
i−1
In other words, the mapping Deξ g(t, e
ξ|Teξ S n ) is an isomorphism of Rn
to the space TeS n ⊂ Rn+1 for every pair (t, e
ξ)ǫ(−r, r) × σ j and hence
ξ
can be considered as a one-to-one linear mapping from Rn into Rn+1
(with range Teξ S n ⊂ Rn+1 ). A simple but crucial observation is that the
regularity C m−k of the mapping Deξ g (Lemma 3.3) yields the regularity
C m−k of the mapping
i−1
h
(t, e
ξ)ǫ(−r, r) × σ j → Deξ g(t, e
ξ)|Teξ S n ǫL (Rn , Rn+1 ).
This is easily seen by considering a chart of S n around e
ξ0j and can
be formally seen by observing that the dependence of the tangent space
Teξ S n on the variable e
ξǫS n is C ∞ while taking the inverse of an invertible
linear mapping is a C ∞ operation. Setting, for (t, e
ξ)ǫ(−r, r) × σ j ,
i−1
h
ξ)
ϕ j (t, e
ξ) = Deξ g(t, e
ξ)|Teξ S n h(t, e
2. A generalization of the Morse Lemma
52
and noting that hǫC m−k (R × Rn+1 , Rn ), we deduce
ϕ j ǫC m−k ((−r, r) × σ j , Rn+1 ).
66
Note from (4.4) that ϕ j (0, e
ξ) = 0 for e
ξǫσ j . On the other hand, by
implict differentiation of the identity g(t, e
ξ j (t)) = 0 for 0 < |t| < r, (the
chain rule applies since g is differentiable with respect to (t, e
ξ) at any
point of (R − {0}) × Rn+1 ),
de
ξj
∂q ej
(t, ξ (t)) + Deξ g(t, e
ξ j (t)) ·
(t) = 0.
∂t
dt
But (de
ξ j /dt)(t)ǫTeξ j (t) S n , since e
ξ j takes its values in S n . Hence
i−1 ∂q
h
de
ξj
(t) = − Deξ g(t, e
ξ j (t))|Teξ j (t) S n
(t, e
ξ j (t)).
dt
∂t
With Lemma 4.2, this yields
de
ξj
1
(t) = − ϕ j (t, e
ξ j (t)), 0 < |t| < r
dt
t
and the relation (4.7) can be rewritten as
de
xj
(t) = −ϕ j (t, e
ξ j (t)) + e
ξ j (t), 0 < |t| < r.
dt
(4.8)
But (cf. Corollary 3.1)
whereas
de
xj
j
(0) = e
ξ0 ,
dt
−ϕ j (0, e
ξ(0)) + e
ξ j (0) = −ϕ j (0, e
ξ0j ) + e
ξ0j = ξ0j .
since ϕ j (0, e
ξ) = 0 for e
ξǫσ j . Thus the identity (4.8) holds for every
tǫ(−r, r). As its right hand side is of class C m−k , we obtain
de
ξ j m−k
ǫC ((−r, r), Rn ).
dt
2.4. Further Regularity Results.
67
53
As a last step, it remains to show that the curves generated by the
functions e
ξ j , 1 ≤ j ≤ 2ν have the same regularity as the functions e
xj
themselves. From an elementary result of differential geometry, it is
sufficient to prove that
de
xj
(t) , 0, tǫ(−r, r).
dt
But this is immediate from (4.8) by observing that e
ξ j(t) and ϕ j (t,
j
j
are orthogonal, since ϕ (t, e
ξ (t))ǫTeξ j (t) S n . Hence
e
ξ j (t))
||
de
x j (t)
|| ≥ ||ξ j (t)|| = 1, tǫ(−r, r)
dt
and the proof is complete.
COMMENT 4.1. When k = 1, the condition (R − N.D.) amounts to
saying that D f (0) is onto. In particular, ν = 1 and the local zero set
of f is made up of exactly one curve of class C m away from the origin
and also C m at the origin: the conclusion is the same as while using the
Implict function theorem.
COMMENT 4.2. Assume now n = 1 and k = 2. The condition (R −
N.D.) is the Morse condition and we know that ν = 0 or ν = 2. The
statement is noting but the weak form of the Morse Lemma.
COMMENT 4.3. In the same direction see the articles by Magnus [24],
Buchner, Marsden and Schecter [5] Szulkin [40] among others. Theorem 4.1 is a particular case of the study made in Rabier [29]. More
generally, the following extension (which, however, has no major application in the nondegenerate cases we shall consider in Chapter 3) does
not requires D f (0) to vanish. Such a result is important in generalizations of the desingularization process we shall describe in Chapter 5.
Theorem 4.2. Let f be a mapping of class C m , m ≥ 1, defined on a 68
neighbouhood of the origin in Rn+1 with values in Rn . Let us set
n0 = n − dim RangeD f (0),
54
2. A generalization of the Morse Lemma
so that 0 ≤ n0 ≤ n and the spaces Ker D f (0) and Rn /Range D f (0) can
be identified with Rn0 +1 and Rn0 respectively. Let π denote the canonical
projection operator from Rn onto Rn /Range D f (0). We assume that



 f (0) = 0


 πD j f (0) = 0, 1 ≤ j ≤ k − 1
for some 1 ≤ k ≤ m and that the mapping
q0 : e
ξǫ Ker D f (0) ≃ Rn0 +1 → q0 (e
ξ) = πDk f (0) · (e
ξ)k ǫRn /RangeD f (0) ≃ Rn0
verifies the condition (R − N.D.). Then the zero set of the mapping consists of a finite number ν ≤ kn0 of lines through the origin in Ker D f (0)
and the local zero set of the mapping f consists of exactly ν curves of
class C m away from the origin and of class C m−k+1 at the origin. These
ν curves are tangent to a different one from among the lines of the zero
of the mapping q0 at the origin.
For a proof, cf. [29]. Theorem 3.2.
2.5 A Generalization of the Strong Version of the
Morse Lemma.
69
To complete this chapter, we shall see that the local zero set of f can be
deduced from the zero set of the polynoimal mapping q(e
ξ) = Dk f (0) ·
(e
ξ)k through an origin-preserving C m−k+1 local diffeomorphism of the
ambient space Rn+1 under some additional assumptions on the lines in
the zero set of q.
Theorem 5.1. Under the additional assumptions that ν ≤ n + 1 and the
ν lines in the zero set of q are linearly independent (a vacuous condition
if ν = 0), there exists an origin-preserving local diffeomorphism φ of
class C m−k+1 in Rn+1 that transforms the local zero set of the mapping q
into the local zero set of the mapping f. Moreover, φ is of class C m away
from the origin and can be taken so that Dφ(0) ≡ I.
Proof. If ν = 0, we can choose φ = I. If ν ≥ 1, the local zero set of f
is the image of the ν functions e
x j (t), 1 ≤ j ≤ ν, |t| < r. These functions
2.5. A Generalization of the Strong Version.....
55
are of class C m−k+1 at the origin and of class C m away from it, with
j
(de
x j /dt)(0) = e
ξ0 , 1 ≤ j ≤ ν.
Let us denote by (·|·) the usual inner product of Rn+1 . We may write
where
ζ j (t),
e
x j (t) = α j (t)e
ξ0j + e
α j (t) = (e
x j (t)|e
ξ0j ).
Therefore, both functions α j and e
ζ j are of class C m−k+1 at the origin 70
and of class C m away from it. Besides,

dα j


 α j (0) = 0ǫR, dt (0) = 1,
(5.1)

ej

 e
ζ j (0) = 0ǫRn+1 , dζ (0) = 0ǫRn+1 .
dt
From (5.1), after shrinking r if necessary, we deduce that the function α j is a C m−k+1 diffeomorphism from (−r, r) to an open interval I j
containing zero. We shall set
(−ρ, ρ) =
ν
\
I j, ρ > 0
j=1
m−k+1 in (−ρ, ρ) and
so that each function α−1
j is well defined, of class C
m
of class C away from the origin,
As the ν vectors e
ξ0j , 1 ≤ j ≤ ν are linearly independent we can find
a bilinear form a(·, ·) on Rn+1 such that
a(e
ξ0i , e
ξ0j ) = δi j (Kronecker delta), 1 ≤ i, j ≤ ν
For ||e
ξ|| < ρ, let us define
φ(e
ξ) = e
ξ+
ν
X
j=1
(5.2)
e
ζ0j◦ α1j ◦ a(e
ξ, e
ξ0j )
clearly, φ is of class C m−k+1 at the origin and of class C m away from it
and further φ(0) = 0, Dφ(0) = I. Therefore φ is an origin preserving
local diffeomorphism of Rn+1 having the desired regularity properties.
2. A generalization of the Morse Lemma
56
Let us show that φ transforms the local zero set of q into the local zero
set of : For |τ| < ρ and 1 ≤ i ≤ ν
j
φ(τe
ξ0i ) = τe
ξ0 +
= τe
ξ0i +
=
71
i
ν
X
j=1
ν
X
j=1
e
ei ej
ζ j (α−1
j (a(τξ0 , ξ0 )))
e
ei ei −1
ζ(α−1
j (τ)δi j ) = τξ0 + ζ (αi (τ))
e
x (α−1
i (τ)).
COMMENT 5.1. Assume k = 2, n = 1. Then the condition (R −
N M.D.) is equivalent to the Morse condition and ν = 0 or ν = 2. In any
case, ν ≤ n + 1 = 2 and, if ν = 2, the two lines are distinct. Thus, they
are linearly independent and Theorem 5.1 coincides with the “strong”
version of the Morse lemma (Theorem 3.1 of Chapter 1).
Remark 5.1. If the diffeomorphism φ is only required to be C 1 at the
origin, the result is true regardless of the number of lines in the zero set
of q(≤ kn because of the condition (R − N.D.)) and without assuming
of course that they are linearly independent. The condition also extends
to mapping from Rn+p into Rn , p ≥ 1. Under this form, there is also
a stronger result when a stronger assumption holds : Assume for every
e
ξǫRn+1 − {0} that the derivative Dq(e
ξ) is onto. Then, the diffeomorphism
φ can be taken so that
f (φ(e
ξ)) = q(e
ξ),
for e
ξ around the origin (see Buchner, Marsden and Schecter [5]). For
k = 1, the assumption reduces to sayong that D f (0) is onto. For k = 2
and n = 1, it reduces to the Morse condition again.
Chapter 3
Applications to Some
Nondegenerate problems
The AIM of this Chapter is to show how to apply the general results of 72
Chapter 2, 2.4, to the one-parameter nonlinear problems introduced in
Chapter 1. Such an application is possible either directly or after a preliminary change of the parameter. We answer such questions as getting
an upper and a nontrivial lower bound for the number of curves, their
regularity and their location in space. We have limited ourselves to the
study of two problems, presented in §§2 and 3 respectively. These examples are drawn from a global synthetic approach developed in Rabier
[31], whose technicalities see to be too tedious to be wholly reproduced
here.
In the first section, we prove a generalization of Theorem 3.2 of
Chapter 1. This generalization is of interest because it shows that when
the results of Chapter 2, §4, are applied to a mapping f which is the
reduced mapping of some problem posed between real Banach spaces,
the assumptions on f are independent of the Lyapunov-Schmidt reduction used for reducing the problems to a finite-dimensional one.
The second section is devoted to problems of bifurcation from the
trivial branch at a multiple characteristic value. As the Morse lemma
was shown to provide the results of Crandall and Rabinnowitz when the
characteristic value is simple (cf. Chapter 1), the use of the extended 73
57
3. Applications to Some Nondegenerate problems
58
version yields again, with various additional improvements, the conclusions of the earlier work by Mc Leod and Sattinger in their study of the
same problem ([23]). Thus, the analysis of bifurcation from the trivial branch at a simple or multiple characteristic value appears to follow
from the same general statement, giving some homogeneity to the presentation.
In the second section, we consider another example, in which no
branch of solutions is known a priori. In the simplest form of the problem, two typical situations are those when the origin is a “turning point”
or a “hystersis point”. These notions are made precise and the structure
of the local zero set is also determined in the presence of a higher order
singularity. In particular, it is shown that bifurcation can be expected in
this case.
3.1 Equivalence of Two Lyapunov-Schmidt Reductions.
Here, we shall define and prove the equivalence of any two LyapunovSchmidt reductions of a given problem. The notion of equivalence is a
key tool for proving a general version of Theorem 3.2 of Chapter 1. The
results of this section are due to W.J. Beyn [42].
e and Y be two real Banach spaces and G : X
e → Y1 a
Let then X
m
mapping of class C , m ≥ 1, satisfying the conditions:
G(0) = 0,
(1.1)
DG(0) is a Fredholm operator with index 1.
(1.2)
74
As in Chapter 1, we shall set
e1 = Ker DG(0),
X
Y2 = Range DG(0),
1
(1.3)
(1.4)
Actually, the mapping G needs only to be defined in a neighbourhood of the origin.
3.1. Equivalence of Two Lyapunov-Schmidt Reductions.
59
e1 has finite dimension
so that Y2 has finite codimension n ≥ 0 and X
n + 1 from the assumption (1.2). Given two topological complements
e2 and Y1 of X
e1 and Y2 respectively, recall that Q1 and Q2 denote the
X
(continuous) projection operators onto Y1 and Y2 respectively and that,
e in the form e
writing e
xǫ X
x=e
x1 + e
x2 , the reduced mapping is defined by
e1 → f (x1 ) = Q1G(e
e
x1 ǫ X
x1 + e
ϕ(e
x1 ))ǫY1 ,
(1.5)
Q2G(e
x1 + e
ϕ(e
x1 )) = 0,
(1.6)
e2 is characterized by
where the mapping e
ϕ with values in the space X
e1 (so that the reduced mapping f
for e
x1 around the origin in the space X
e1 ). In
in (1.5) is actually defined in a neighbourhood of the origin in X
Theorem 1.1 below, we show that the mappings G(e
x) and f (e
x1 )+DG(0)·
e
e
x2 differ only in the context of “changes of variables” in the spaces X
and Y. More precisely,
e of the origin in the space X
e 75
Theorem 1.1. There is a neighbourhood U
and
e I som(Y)),
(i) a mapping τǫC m−1 (U,
e X)
e with De
(ii) an origin-preserving diffeomorphism e
ρǫC m (U,
ρ(0) =
IXe such that
e
for every e
xǫ U.
e ρ(e
e
τ(X)G(e
x)) = f (e
x1 ) + DG(0)e
x2 ,
e set
Proof. For e
x around the origin in the space X,
i−1
h
e x) = e
x).
R(e
x1 + DG(0)|Xe2 Q2 G(e
(1.7)
(1.8)
e is of class C m and R(0)
e = 0, DR(0)
e = I e so
Then, the mapping R
X
e is on origin-preserving local C m -diffeomorphism of the space X.
e
that R
Besides, from (1.8), we have
i
h
e x) = DG(0)| e −1 Q2G(e
x)
Q2 R(e
X2
3. Applications to Some Nondegenerate problems
60
and hence
e x) = Q2G(e
DG(0)Q2 R(e
x).
e−1 , it follows that
Setting e
ρ=R
Q2G(e
ρ(e
x)) = DG(0)e
x2 ,
(1.9)
e of the origin in X.
e Note from (1.8) that
for e
x in some neighbourhood U
e
ρ is of the form
e
ρ(e
x) = e
x1 + e
ϕ(e
x),
(1.10)
76
e X
e2 ). In particular, putting e
with e
ϕǫC m (U,
x=e
x1 in (1.9) (i.e. e
x2 = 0), we
find that e
ϕ(e
x1 ) is characterized by (1.6), which agrees with our notation.
Besides
ρ(e
x1 ) = Q1 G(e
ρ(e
x1 )).
(1.11)
e is convex.
It is not restrictive to assume that the neighbourhood U
Let us then define
τ(e
x) = Q1 + τ12 (e
x)Q2 + Q2
(1.12)
x)
where τ(e
12 ǫL (Y2 , Y1 ) is given by
τ12 (e
x) = −
"Z
1
0
#h
i−1
Q1 DG(e
ρ(e
x1 + se
x2 ))Dex2 e
ρ(e
x1 + se
x2 )dx DG(0)|Xe2 .
(1.13)
e
Clearly,
and τ(0) = IY so that, after shrinking U
e
if necessary, τ(e
x) is an isomorphism of Y for every e
xǫ U. Now, we have
e L (Y))
τǫC m−1 (U,
τ(e
x)G(e
ρ(e
x)) = Q1G(e
ρ(e
x)) + τ12 (e
x)Q2G(e
ρ(e
x)) + Q2G(e
ρ(e
x)) =
(1.14)
= Q1G(e
ρ(e
x)) + τ12 (e
x)DG(0)e
x2 + DG(0)e
x2 .
From (1.11) and the Taylor formula, we have
Q1G(e
ρ(e
x)) = f (e
x1 ) +
Z
0
1
Q1 DG(e
ρ(e
x1 + se
x2 ))Dex2 e
ρ(e
x1 + se
x2 ) · e
x2 dx,
and the assertion follows from (1.13) and (1.14).
3.1. Equivalence of Two Lyapunov-Schmidt Reductions.
61
e−1 and if we denote by R
e1 and R
e2 the projections
By definition, e
ρ=R
e
e
e
of the mapping R onto the spaces X1 and X2 respectively, we can then
write for e
x around the origin,
i
h
e1 (e
e2 .
G(e
x) = (τ(e
x))−1 f (R
(1.15)
x)) + DG(0)R
Let us now consider a second Lyapunov-Schmidt reductions cor- 77
e′ of the complement of X
e1 and a write
responding to a new choice X
2
e1 and e
e′ . We obtain a new reduced equation
e
x=e
x′1 + e
x′2 with e
x′1 ǫ X
x′2 ǫ X
2
′
fˆ = fˆ(e
x1 ) with which Theorem 1.1 applies. Together with (1.15), we
e of the origin in the space X
e and
see that there is a neighbourhood U
e I som(Y)),
(i) a mapping τǫC m−1 (U,
e X)
e with onto the
(ii) an origin-preserving diffeomorphism e
ρǫC m (U,
e
e
spaces X1 and X2 respectively, one has
fˆ(e
x′1 ) + DG(0)e
x′2 = τ(e
x) f (e
ρ1 (e
x)) + DG(0)e
ρ(e
x) ,
(1.16)
e The linear mapping τ(e
for every e
xǫ U.
x) has a matrix representation of the form
"
#
τ̂11 (e
x) τ̂12 (e
x)
τ(e
x) =
,
τ21 (e
x) τ22 (e
x)
e (Yi , Ŷ1 )), i = 1, 2 and τ2i ǫC m−1 (U,
e (Yi , Y2 )), i =
where τ̂1i ǫC m−1 (U,
1, 2. In this notation, relation (1.16) can be rewritten as
fˆ(e
x′1 ) = τ̂11 (e
x) f (e
ρ1 (e
x)) + τ̂12 (e
x)DG(0)e
ρ2 (e
x),
(1.17)
DG(0)e
x′2
(1.18)
= τ21 (e
x) f (e
ρ1 (e
x)) + τ22 (e
x)DG(0)e
ρ2 (e
x),
e
for every e
xǫ U.
Assume first that m ≥ 2, so that the mappings τ21 and τ22 are of 78
class C 1 at least. By differentiating (1.18) and noting that De
ρ(0) = Iex ,
we obtain
e′ = τ21 (0)D f (0) + τ22 (0)DG(0)P
e2 ,
DG(0)P
2
(1.19)
3. Applications to Some Nondegenerate problems
62
e2 and P
e′ denote the projection operators along X
e1 and onto the
where P
2
′
e2 and X
e respectively. When m = 1, the mappings τ21 and τ22
spaces X
2
0
are of class C only. Nevertheless, from the fact that the mappings f · e
ρ1
and DG(0) ·e
ρ2 are of class C 1 and vanish at the origin, it is immediately
verified that both the mapping
e
x2 → τ21 (e
x2 ) f (e
ρ1 (e
x)),
and
e
x2 → τ22 (e
x)DG(0)e
ρ2 (e
x),
are differntiable at the origin and that formula (1.19) still holds. Besides,
D f (0) = 0 (cf, Chapter 1, §2) and hence
In particular,
e′ = τ22 (0)DG(0)P
e2 .
DG(0)P
2
e′ | e = τ22 (0)DG(0)| e .
DG(0)P
2 X2
X2
e2 , Y2 ), we get
As DG(0)|Xe2 ǫI som(X
79
i−1
h
e′ | e .
τ22 (0) = DG(0)|Xe2 DG(0)P
2 X2
(1.20)
e′ = DG(0)| e′ . Indeed, as X
e2
At this stage, observe that DG(0)P
X2
2
e′ are two topological complements 2 of the space X
e1 , it suffices
and X
2
′
e | e is onr-to-one and onto and use the Open mapping
to notice that P
2 X2
theorem. Therefore, writing (1.20) in the form
i
i−1 h
h
DG(0)|Xe2 P′2 |Xe2 ,
τ22 (0) = DG(0)|Xe2
e
it follows that τ22 (0)ǫ Isom (Y2 ). After shrinking the neighbourhood U
e
if necessary, we may then suppose that τ22 (e
x)ǫ Isom (Y2 ) for every e
xǫ U.
e1 (i.e. e
Now, let us take e
xǫ X
x = e
x1 = e
x′1 ) in (1.17) and (1.18) to
obtain
fˆ(e
x1 ) = τ̂11 (e
x1 ) f (e
ρ2 (e
x1 )) + τ̂12 (e
x1 )DG(0)e
ρ2 (e
x1 ),
2
(1.21)
e2 and X
e′ are closed in X
e and hence are Banach spaces by themselves.
In particular, X
2
3.1. Equivalence of Two Lyapunov-Schmidt Reductions.
63
0 = τ11 (e
x1 ) f (e
ρ1 (e
x1 )) + τ22 (e
x1 )DG(0)e
ρ2 (e
x1 ),
(1.22)
From (1.22) and the above comments, we see that
DG(0)e
ρ2 (e
x1 ) = −(τ22 (e
x1 ))−1 τ21 (e
x1 ) f (e
ρ1 (e
x1 )).
With this relation, (1.21) becomes
h
i
fˆ(e
x1 ) = τ̂11 (e
x1 ) − τ̂12 (e
x1 )(τ22 (e
x1 ))−1 τ21 (e
x1 ) f (e
ρ1 (e
x1 )).
Let us set
τ̂(e
x1 ) = τˆ11 (e
x1 ) − τ̂12 (e
x1 )(τ22 (e
x1 ))−1 τ21 (e
x1 ).
Clearly, τ̂1 is of class C m−1 on a neighbourhood of the origin in the 80
e1 with values in the space L (Y1 , Ŷ1 ). In addition, τ̂1 (0)ǫ isom
space X
ˆ
(Y1 , Y1 ). Indeed, since Y1 and Ŷ1 have the same dimension n, it suffices
to show that τ̂1 (0) is one-to-one. Let Y1 ǫY1 be such that τ̂1 (0)y1 = 0,
namely
τ̂11 (0)y1 − τ̂12 (0)(τ22 (0))−1 τ21 (0)y1 = 0
If so, we seet that
#
"
#"
y1
τ̂11 (0) τ̂12 (0)
=0
τ21 (0) τ22 (0) − (τ22 (0))−1 τ21 (0)y1
and hence y1 = 0. It follows that τ̂1 (e
x)ǫ Isom (Y1 , Ŷ1 ) for e
x1 near the
e1 and we have proved
origin in the space X
e1 of the origin in the space
Theorem 1.2. There is a neighbourhood U
e
X1 and
e1 , I som(Y1 , Ŷ1 )),
(i) a mapping τ̂1 ǫC m−1 (U
(ii) an origin-preserving
De
ρ1 (0) = IXe′ such that
1
diffeomorphism
e1 , X
e1 )
e
ρ1 ǫC m (U
fˆ(e
x1 ) = τ̂1 (e
x1 ) f (e
ρ1 (e
x1 )),
e1 .
for every e
x1 ǫ U
with
(1.23)
64
3. Applications to Some Nondegenerate problems
Remark 1.1. It is customary to summarize Theorem 1.2 by saying that
any two Lyapunov-Schmidt reductions are equivalent. For our purpose,
this property is important because of Corollary 1.1 below.
81
Corollary 1.2. Assume that there is an integer k ≤ m such that
D j f (0) = 0, 0 ≤ j ≤ k − 1,
and the mapping
e1 → Dk f (0) · (e
e
x1 ǫ X
x1 )k ǫY1 ,
(1.24)
verifies the condition (R − N.D.). Then, one has
D j fˆ(0) = 0, 0 ≤ j ≤ k − 1,
and the mapping
x1 ǫX1 → Dk fˆ(0) · (e
x1 )k ǫ Ŷ1 ,
(1.25)
verifies the condition (R − N.D.).
Proof. With the notation of Theorem 1.2, note that in view of De
ρ1 (0) =
IXe1 that
D j( f • e
ρ1 )(0) = D j f (0), 0 ≤ j ≤ k.
(1.26)
Assume first k ≤ m − 1, so that, for every 0 ≤ j ≤ k, D j fˆ(0) involves
the derivatives of order ≤ k of the mappings τ̂1 and f • ρ1 at the origin
as it follows from (1.23). With (1.26), a simple calculation provides
D j fˆ(0) = 0, 0 ≤ j ≤ k − 1,
Dk fˆ(0) = τ̂1 (0) Dk f (0).
82
(1.27)
(1.28)
If k = m, there is a slight difficulty in applying the same method
since the mapping τ̂1 is not m times differentiable. However, it is easy
to obtain Dm−1 fˆ(e
x1 ) in terms of the derivatives of order ≤ m − 1 of the
mappings τ̂1 and ( f • e
ρ1 ). We leave it to the reader to check that each
term of the expression is differentiable at the origin and the relations
(1.27) and (1.28) still hold. In particular, as τ̂1 (0)ǫI som(Y1 , Ŷ1 ), it is
an obvious consequence of (1.27) that the mapping (1.25) verifies the
condition (R − N.D.) as soon as the mapping (1.24) does.
3.2. Application to Problems of Bifurction.....
65
Remark 1.2. From Corollary 1.2, assuming that the first nonzero derivative at the origin of the reduced mapping f verifies the condition (R −
e2 and Y1 and hence is
N.D.) is independent of the choice of the spaces X
an intrinsic property of the mapping G.
3.2 Application to Problems of Bifurction from the
Trivial Branch at a Multiple Characteristic
Value
According to our definitions of Chapter 1, given a real Banach space X,
we are interested in finding the solutions (µ, x) near the origin of R × X
of an equation of the form
G(µ, x) = (I − (λ0 + µ)L)x + Γ(µ, x) = 0,
(2.1)
where LǫL (X) is compact and λ0 is a multiple characteristic value of L,
i.e.,
n = dimKer(I − λ0 L) ≥ 2.
Recall the notation (cf. Chapter 1):
X1 = Ker(I − λ0 L),
Y2 = Range(I − λ0 L),
(2.2)
(2.3)
and X2 and Y1 are arbitrary topological complements of X1 and Y2 re- 83
spectively. If so,
dimX1 = dimY1 = n.
(2.4)
Also, Q1 and Q2 denote the (continuous) projection operators onto
the spaces Y1 and Y2 . The nonlinear operator Γ verifies conditions
Γ(µ, 0) = 0,
(2.5)
j
(so that Dµ Γ(0) = 0 for every j ≥ 0) and
D x Γ(0) = Dµ D x Γ(0) = 0.
(2.6)
66
3. Applications to Some Nondegenerate problems
In what follows, we shall assume that there is an integer k ≤ m (and
necessarily ≥ 2) such that
Q1 D xj Γ(0) = 0, 0 ≤ j ≤ k − 1,
Q1 Dkx Γ(0)|(X1 )k , 0,
(2.7)
As we saw in Chapter 1, §2, the problem amounts to finding the
local zero set of the reduced equation
µ
µ
f (µ, x) = − Q1 x − Q1 ϕ(µ, x) + Q1 Γ(µ, x + ϕ(µ, x)) = 0. (2.8)
λ0
λ0
84
for (µ, x)ǫR×X1 , where the mapping ϕ with values in X2 is characterized
by the properties



Q2G(µ, x + ϕ(µ, x)) = 0,
(2.9)


ϕ(0) = 0,
for (µ, x) near the origin of R × X1 .
Remark 2.1. To avoid repetition, we shall not state the corresponding
properties of the local zero set of the mapping G. The reader can check
without any difficulty that all the results about the local zero set of the
reduced mapping f (number of curves, regularity, location in the space)
remain valid ad concerns the local zero set of G.
As already seen in Chapter 1 in a general setting, the derivatives
D f (0) and Dϕ(0) vanish. An elementary calculation thus provides
D2 f (0) · (µ, x)2 = −
2µ
Q1 x + Q1 D2x Γ(0) · (x)2 ǫY1 ,
λ0
(2.10)
for (µ, x)ǫR × X1 . Note, in particular, that D2 f (0) = 0 if f X1 ⊂ Y2 and
k ≥ 3 simulaneously.
THE CASE k = 2
If k = 2, one has D2 f (0) , 0 and the mapping (2.10) must verify the
condition (R − N.D.). An implict necessary condition for this is
X = X1 ⊕ Y2 (= Ker(I − λ0 L) ⊕ Range(I − λ0 L)).
(2.11)
3.2. Application to Problems of Bifurction.....
67
Recall that the necessity of such a decomposition was already noticed when n = 1 in Chapter 1. In other words, it is necessary that the
algebraic multiplicity of λ0 equals its geometric multiplicity.
Indeed, assume X1 ∩ Y2 , {0}. The line {(µ, 0), µǫR} (trivial banch) 85
is clearly in the zero set of the mapping (2.10) and its derivative at any
point of this line is
(µ′ , x′ )ǫR × X1 → −
2µ
Q1 x′ ǫY1 ,
λ0
(2.12)
whose range is that of Q1 |X1 . But, from the assumption X1 ∩ Y2 , {0},
we deduce that dim Ker Q1 |X1 ≧ 1. Hence dim Range Q1 |X1 ≤ n − 1 and
the mapping (2.12) is not onto.
Now, from the results of §1, we see that the validity (or the failure) of
the condition (R−N.D.) is independent of the choice of the complements
X2 of X1 and Y1 of Y2 . From (2.11), we may choose X1 = Y1 and the
mapping (2.10) becomes
D2 f (0) · (µ, x)2 = −
2µx
+ Q1 D2x Γ(0) · (x)2 ǫX1 ,
λ0
(2.13)
for (µ, x)ǫR × X1 . Its derivative at any point (µ, x)ǫR × X1 is the linear
mapping
!
µ ′ µ′
2
′
(µ , x )ǫR × X1 → 2 − x − x + Q1 D x Γ(0) · (x, x ) ǫX1 . (2.14)
λ0
λ0
′
′
It is clearly onto at each point of the form (µ, 0) with µ , 0. Whether
or not it is onto at the other nonzero solutions of the equation D2 f (0) ·
(µ, x)2 = 0 has to be checked in each particular problem separately.
Simple finite-dimensional examples show that this hypothesis is quite
realistic.
Remark 2.2. It is not restrictive to limit ourselves to finite dimensional 86
examples. Indeed, the mapping (2.13) involves finite dimensional terms
only; in particualr, the term Q1 D2x Γ(0) · (x)2 is nothing but the second
derivative with respect to x of the mapping Q1 Γ|R×X1 at the origin.
3. Applications to Some Nondegenerate problems
68
When the results of Chapter 2, §4 hold, the largest number ν of
curves (which are of class C m−1 at the origin and of class C m away from
it) is 2n since k = 2. As k is even, ν must be even too (cf. Chapter 2, §1).
The trivial branch being in the local zero set of f . existence of a second
curve is then ensured, namely, bifurcation does occur.
COMMENT 2.1. When n = 1 and Q1 D2x Γ(0)|(X1 )2 , 0 it is immediate
that the nontrivial curve in the zero set of f has a nonvertical tangent
at the origin. Thereofre, bifurcation occurs transcritically which means
that the nontrivial curve in question is located on both sides of the axis
{0} × X1 in R × X1 . The situation is different when n ≥ 2 and there may
be curves in the local zero set of f which are tangent to some lines of
the hyperplane {0} × X1 of R × X1 at the origin, as in the example below.
However, this cannot happen if the hypothesis Q1 D2x Γ(0)|(X1 )2 , 0 is
replaced by the stronger one Q1 D2x Γ(0) · (x)2 , 0 for every xǫX1 − {0}
(note that the two assumptions are equivalent when n = 1).
87
EXAMPLE . Take X = R2 , L = I and λ0 = 1. Thus, X1 = X = R2 ,
Q1 = I and X2 = {0}. Writing x = (x1 , x2 ), x1 , x2 ǫR and with
#
x22 + x31
,
Γ(µ, x) = Γ(x) =
x1 x2
"
we find
#
−µx1 + x22
.
D f (0) · (µ, x) = 2
−µx2 + x1 x2
2
2
"
This mapping verifies the condition (R − N.D.) and its zero set is
made up of the 4 lines: R(1, 0, 0) (trivial branch), R(0, 1, 0), R(1, 1, 1)
and R(1, 1, −1). The zero set of f q
is made up of the 4 curves:
q (µ, 0, 0)
(trivial branch), (x21 , x1 , 0), (x1 , x1 ,
x21 − x31 ) and (x1 , x1 , −
x21 − x31 ).
The curve (x21 , x1 , 0) is tangent to the plane {0} × X1 at the origin and
located in the half-space µ ≥ 0 (cf. Fig. 2.1)
3.2. Application to Problems of Bifurction.....
69
3
2
2
1
3
1
2
Figure 2.1:
COMMENT 2.2. The same picture as above describes the local zero set 88
of G: the only modification consists in replacing the hyperplane {0} × X1
of R × X1 by the hyperplane {0} × X in R × X.
THE CASE k ≥ 3.
If k ≥ 3, the mapping (2.10) is
D2 f (0) · (µ, x)2 = −
2µ
Q1 x,
λ0
(2.15)
because Q1 D2x Γ(0) = 0. When X1 1 Y2 (which can be expected in most
of the applications), one has D2 f (0) , 0 so that the mapping (2.15)
should verify the condition (R − N.D.). Unfortunately, this is never the
case when n ≥ 2 (as assumed throughout this section). To see this, note
that the local zero set of the mapping (2.15) contains the pair (0, x) for
any xǫX1 . Its derivative at such a point is given by
(µ′ , x′ )ǫR × X1 → −
2µ′
Q1 xǫY1 .
λ0
Its range is the space RQ1 x of dimension ≤ 1 and it cannot be onto
for n ≥ 2 (note that it is onto when n = 1 and X = X1 ⊕ Y2 so that there
is no contradiction with the results of Chapter 1).
3. Applications to Some Nondegenerate problems
70
However, it is possible to overcome the difficulty by performing a
em change of scale in our initial problem. The idea is as follows. For
any odd integer p, the mapping
ηǫR → ηρ ǫR,
89
is a C ∞ homeomorphism. One can then wonder whether there is a “suitable” choice of the odd integer ρ ≥ 3 such that, setting µ = η p , the
results of Chapter 2, §4 apply to the problem: Find (η, x) around the
origin of R × X such that
G(η p , x) = (I − (λ0 + η p )L)x + Γ(η p , x) = 0.
(2.16)
The Lyapunov-Schmidt reduction of this new problem yields a reduced equation
g(η, x) = 0,
for (η, x) around the origin in R × X1 , where the mapping is easily seen
to be
g(η, x) = f (η p , x),
the mapping f being the reduced mapping in (2.8) of the problem (2.1).
In particular, the mapping which is found through the Implict function
theorem in the Lyapunov-Schmidt reduction is exactly ϕ(η p , x) where ϕ
is characterized by (2.9) and the solutions to the problem (2.1) are the
pairs
{(η p , x + ϕ(η p , x))ǫR × X, (η, x)ǫR × X1 , g(η, x) = 0}.
(2.17)
Intuitively, the choice of p can be made by examinin the function
g(η, x) = −
90
ηp
ηp
Q1 x − Q1 ϕ(η p , x) + Q1 Γ(η p , x + ϕ(η p , x)).
λ0
λ0
(2.18)
As Dϕ(0) = 0, the leading term in the expression −(η p /λ0 )Q1 x −
x) is −(η p /λ0 )Q1 x. On the other hand, in analogy with
the case k = 2, it might be desirable that the term Q1 Dkx Γ(0) · (x)k as well
as the first nonzero derivative of the term −(η p /λ0 )Q1 x at the origin be in
the expression of the first nonzero derivative of g at the origin. The later
(η p /λ0 )Q1 ϕ(η p ,
3.2. Application to Problems of Bifurction.....
71
is of order p + 1 whereas the term Dkx Γ(0) · (x)k does not appear before
differentiatinf k times. A “good” relation thus seems to be p + 1 = k,
namely p = k − 1.
This turns out be the right choice of p, which can actually be found
after eliminating all the other values on the basis of natural mathematical
requirements instead of using the above intuitive arguments. Of course,
a mathematical method for finding the proper change of parameter may
have some importance in problems in which there is no a priori guess of
what it should be. Incidentally, one can also provide a complete justification of the use of Newton diagrams in the change of scale as is done,
for instance, in Sattinger [34]. However, the study is technical and too
long to be presented here (cf. Rabier [31]).
As we have assumed that p is odd for the reasons explained above,
we must suppose that k is even. A similar method will be analysed later
on when k is odd.
The case k even: We need to find the first nonzero derivative of the
mapping of (2.18) at the origin in R × X1 when p = k − 1. We can
already guess that it will be of order k but we have to determine its 91
explict expression in terms of the data. With this aim, we first prove
Lemma 2.1. Around the origin in R × X. one has
Q1 Γ(ηk−1 , x) =
1
Q1 Dkx Γ(0) · (x)k + ◦(|η| + |||x|||)k ),
k!
where ||| · ||| denote the norm of the space X.
Proof. From the assumption k ≥ 3, DΓ(0) = 0 and Q1 D2 Γ(0) = 0, since
we already saw that D2µ Γ(0) = 0, Dµ D x Γ(0) = 0 (cf. (2.5) - (2.6)). Thus,
the Taylor formula about the origin yields
Q1 Γ(µ, x) =
k
X
1
Q1 D j Γ(0).(µ, x) j + 0(|µ| + |||x|||)k ),
j!
j=3
For any j with 3 ≤ j ≤ k, one has
!
j
X
j j−i
j−i
Q1 D Γ(0)(µ, x) =
µ Q1 Dµ Dix Γ(0) · (x)i .
i
i=0
j
j
3. Applications to Some Nondegenerate problems
72
j
But, from (2.5) Dµ Γ(0) = 0 for every j and the index i actually runs
over the set {1, · · · , j}. Also, for j < k, it follows, from (2.7), that i runs
over the set 1, · · · , j − 1 only. To sum up,
j−1
X 1 X
1
j
j−i
Q1 Dkx Γ(0) · (x)k +
(i )µ j−i Q1 Dµ
k!
j!
i=1
j=3
k
Q1 Γ(µ, x) =
Dix Γ(0) · (x)i + o((|µ| + |||x|)k ).
Replacing µ by ηk−1 , we shall see that each term in the sum is of
order o(|η| + |||x|||)k ), except for (1/k!)Q1 Dkx Γ(0) · (x)k . This is obvious
as concerns the remainder which becomes o((|η|k−1 + |||x|||)k ) and, for
3 ≤ j ≤ k and 1 ≤ i ≤ j − 1, each term in the sum is of order
0(|η|(k−1)( j−i) |||x|||i ).
92
Replacing |η| and ||x|| by |η| + |||x|||, it is a fortiori of order
0((|η| + |||x|||)(k−1)( j−i)+1 ).
As j − i ≥ 1 and j − i = 1 if and only if i = j − 1, we have, for
i = j−1
(k − 1) + ( j − 1) ≥ k + 1
ecause j ≥ 3. On the other hand, if j − i ≥ 2 and since i ≥ 1
(k − 1)( j − i) + i ≥ 2(k − 1) + 1 = 2k − 1 ≥ k + 1.
In any case, each term is of order
0((|η| + |||x|||)k+1 ),
which completes the proof.
Proposition 2.1. The first nonzero derivative of the mapping
(η, x)ǫR × X1 → g(η, x) = f (ηk−1 , x) = −
ηk−1
ηk−1
Q1 x −
Q1 ϕ(ηk−1 , x)
λ0
λ0
(2.19)
3.2. Application to Problems of Bifurction.....
73
+ Q1 Γ(ηk−1 , x + ϕ(ηk−1 , x))ǫY1
at the origin is of order k and its value at the point (η, x)ǫR×X1 repeated
k times (which determines it completely) is
Dk g(0) · (η, x)k = −
k! k−1
η Q1 x + Q1 Dkx Γ(0) · (x)k .
λ0
(2.20)
Proof. As Dϕ(0) = 0, one has
ϕ(µ, x) = o(|µ| + |||x|||),
around the origin. In particular, replacing µ by ηk−1
ϕ(ηk−1 , x) = o(|η| + |||x||),
93
(2.21)
so that the term (ηk−1 /λ0 )Q1 ϕ(ηk−1 , x) is of order
(ηk−1 /λ0 )Q1 ϕ(ηk−1 , x) = 0(|η|k−1 o(|η| + |||x|||)) = o(|η| + |||x|||)k ). (2.22)
Next, from Lemma 2.1,
1
Q1 Dkx Γ(0) · (x + ϕ(ηk−1 , x))k
k!
+ o((|η| + |||x + ϕ(ηk−1 , x)|||)k ).
(2.23)
Q1 Γ(ηk−1 , x + ϕ(ηk−1 , x)) =
But
|η| + |||x + ϕ(ηk−1 , x)||| = 0(|η| + |||x||| + |||ϕ(ηk−1 , x)|||).
Using (2.21), we see that
o(|η| + |||x||| + |||ϕ(ηk−1 , x)|||) = o(|η| + |||x||| + o(|η| + |||x|||)) = o(|η| + |||x|||).
Thus, the remainder in (2.23) is
o(|η| + |||x + ϕ(ηk−1 , x)|||)k ) = o([0(|η| + |||x|||)]k ) = o((|η| + ||x||)k ),
so that
Q1 Γ(ηk−1 , x+ϕ(ηk−1 , x)) =
1
Q1 Dkx Γ(0)·(x+ϕ(ηk−1 , x))k +o((|η|+|||x|||)k ).
k!
(2.24)
3. Applications to Some Nondegenerate problems
74
Now, from (2.21) again and due to the k-linearity of the derivative
Dkx Γ(0), we deduce
Q1 Dkx Γ(0) · (x + ϕ(ηk−1 , x))k = Q1 Dkx Γ(0) · (x)k + o(|η| + |||x||)k ).
94
By putting this expression in (2.24) and together with (2.22), the
mapping g (2.19) can be rewritten as
g(η, x) = −
ηk−1
1
Q1 x + Q1 Dkx Γ(0) · (x)k + o((|η| + |||x|||)k ).
λ0
k!
(2.25)
This relation shows that the first k − 1 derivatives of g vanish at the
origin. Besides, from Taylor’s formula, we have
g(η, x) =
1 k
D g(0) · (η, x)k + o(|η| + |||x|||)k ),
k!
which, on comparing with (2.25), yields
Dk g(0) · (η, x)k = −
k! k−1
η Q1 x + Q1 Dkx Γ(0) · (x)k .
λ0
Remark 2.3. We leave it to the reader to check that the same result holds
if we replace conditions (2.7) by
D xj Γ(0)|(X1 ) j = 0, 0 ≤ j ≤ k − 1,
Q1 Dkx Γ(0)|(X1 )k , 0.
(2.26)
More generally, it still holds under some combination of the hypotheses (2.7) and (2.26): Denoting by k the order of the first non-zero
derivative with respect to x of the mapping Q1 Γ|R×X1 at the origin (suppose to be finite), set
j
k1 = min{0 ≤ j ≤ k, Q1 D x Γ(0) , 0},
(2.27)
k = min{0 ≤ j ≤
(2.28)
1
k, D xj Γ(0)|(X1 ) j
, 0},
3.2. Application to Problems of Bifurction.....
95
75
so that 2 ≤ k1 , k1 ≤ k. Then, it can be shown (cf. [31]) that Proposition
2.7 remains true when
k ≤ k1 + k1 − 2.
(2.29)
Note that k = k1 in our assumptions whereas k = k1 under the hypotheses (2.26), two cases when the criterion (2.29) is satisfied.
When the mapping Dk g(0) · (η, x)k in (2.20) verifies the condition
(R − N.D.), the results of Chapter 2, §4 give the structure of the local
zero set of g (2.19). We shall derive the structure of the local zero set of
f but, first observe that
Proposition 2.2. The mapping Dk g(0) · (η, x)k in (2.20) can verify the
condition (R − N.D.) only if the following two implict conditions are
fulfilled:
(i) X = X1 ⊕ Y2 (= Ker(I − λ0 L) ⊕ Range(I − λ0 L))3 ,
(ii) Q1 Dkx Γ(0) · (x)k , 0 for every xǫX1 − {0}.
Proof. (i) Assume thatg X1 ∩ Y2 = {0}. The trivial branch {(η, 0); ηǫR}
is in the zero set Dk g(0) · (η, x)k and its derivative at any of its nonzero
points is
k!
(η′ , x′ )ǫR × X1 → − ηk−1 Q1 x′ ǫY1 ,
λ0
whose range is Range Q1 |X1 ( Y1 since Ker Q1 |X1 , {0} from the assumption X1 ∩ Y2 , {0}.
96
k
k
(ii) Assume there is a nonzero xǫX1 such that Q1 D x Γ(0) · (x) = 0.
Then, the line R(0, x) is in the zero set of Q1 Dk g(0) · (η, x)k and its
derivative at (0, x) is
(η′ , x′ )ǫR × X1 → kQ1 Dkx Γ(0) · (xk−1 , x′ )ǫY1 ,
whose null-space contains the trivial branch and the line R(0, x) : its
range is then at most (n − 1)-dimensional.
3
Namely, the algebraic muliplicity of λ0 equals its geometric multiplicity.
76
3. Applications to Some Nondegenerate problems
COMMENT 2.3. Assume that the mapping Dk g(0) · (η, x)k (1.20) verifies the condition (R − N.D.). From our analysis of §1, this assumption
is independent of the choice of the complements X2 and Y1 of X1 and Y2
respectively. In the applications, it follows from Proposition 2.2 that we
can take X1 = Y1 so that the mapping g becomes (note that Q1 ϕ = 0 in
this case)
g(η, x) = −
ηk−1
x + Q1 Γ(ηk−1 , x + ϕ(ηk−1 , x))
λ0
(2.30)
and one has
Dk g(0) · (η, x)k = −k!
ηk−1
x + Q1 Dkx Γ(0) · (x)k ,
λ0
for every (η, x)ǫR × X1 .
97
COMMENT 2.4. From (ii) of Proposition 2.2, none of the lines of the
zero set of Dk g(0)·(η, x)k is “vertical” i.e. lies on the hyperplane {0}× X1
of R × X1 . In other words, if t → (η(t), x(t)) is any curve in the local zero
set of g, one has
dη
(0) , 0.
dt
(2.32)
COMMENT 2.5. The trivial branch is in the zero set of the mapping
Dk g(0) · (η, x)k (and also in the local zero set of g since ϕ(µ, 0) = 0; cf.
Chapter 1). As k is even, we know that the number ν of lines must be
evern too (≤ kn ). Thus ν ≥ 2 and bifurcation occurs (as in the case k =
2). In view (2.32), each curve is located on both sides of the hyperplane
{0} × X1 (i.e. bifurcation occurs transcritically).
3.2. Application to Problems of Bifurction.....
77
2
3
1
3
2
1
Figure 2.2: Local zero set of g.
We now pass to the description of the local zero set of f : it is made
up of the ν curves
t → (µ(t), x(t)),
(2.33)
where µ(t) = ηk−1 (t) and t → (η(t), x(t)) is one of the ν curves in the local 98
zero set of g, hence of class C m−k+1 at the origin and of class C m away
from it. Of course, the ν curves in (2.33) are distinct sicne the mapping
η → ηk−1 is a homeomorphism. Also, it is clear that (dµ/dt)(0) = 0 so
that
d
dx
(µ(t), x(t))|t=0 = (0, (0)).
(2.34)
dt
dt
But, for each curve (η(t), x(t)) in the local zero set of g which is
distinct from the trivial branch, we must have
dx
(0) , 0,
dt
because different curves in the local zero set of g have different tangents
at the origin. Thus, for each curve (µ(t), x(t)) in the local zero set of f
which is distinct from the trivial branch, one has
dx
d
(µ(t), x(t))|t=0 = (0, (0)) , 0.
dt
dt
As a result and further since the functions µ(t) = ηk−1 (t) and x(t)
are of class C m−k+1 at the origin and of class C m away from it, we may
deduce that the nontrivial curves in the local zero set of f are also of
3. Applications to Some Nondegenerate problems
78
99
class C m−k+1 at the origin and of class C m away from it. Besides, due
to (2.34), they are tangent to the hyperplane {0} × X1 of R × X1 at the
origin. Finally, as the sign of µ(t) = ηk−1 (t) changes as that η(t) does,
bifurction remains transcritical (cf. Comment 2.5) as in the case when
k = 2 and the condition Q1 D2x Γ(0) · (x) , 0 for every xǫX1 − {0} holds
(cf. Comment 2.1)4 .
2
1
3
3
1
2
Figure 2.3: Local zero set of f (k even)
Remark 2.4. It is possible for several curves in the local zero set of f
to have the same tangent at the origin: This will happen if and only if
several lines in the zero set of the mapping Dk g(0) · (η, x)k (2.20) have
the same projection onto X1 (along the η-axis). If this is the case, that is
one more reason for the assumptions of Chapter 2, §4 to fail when the
parameter µ is unchanged, for, when the curves are found through Theorem 4.1 of Chapter 2, they must have different tangents at the origin.
100
The case k odd: When k is odd, the mappping η → ηk−1 is no longer
a homeomorphism. However, by performing the change µ = ηk−1 , we
shall find all solutions in the local zero set of f associated with µ ≥ 0. In
order to get the solutions associted with µ ≤ 0, it suffices to perform the
change µ = −ηk−1 .
The method is quite similar and we shall set
gσ (η, x) = f (σηk−1 , x) = −
4
σηk−1
σηk−1
Q1 x −
Q1 ϕ(σηk−1 , x)+
λ0
λ0
Recall however that this condition is not required when k = 2.
3.2. Application to Problems of Bifurction.....
+ Q1 Γ(σηk−1 , x + ϕ(σηk−1 , x)),
79
(2.35)
where σ = ±1. Arguing as in Proposition 2.1, we find that the first
nonzero derivative of the mapping gσ at the origin is of order k with
Dk gσ (0) · (η, x)k = −
σk! k−1
η Q1 x + Q1 Dkx Γ(0) · (x)k ǫY1 ,
λ0
(2.36)
for every (η, x)ǫR × X1 . Again, two implict condition for the results of
Chapter 2, §4 to be available are
X = Ker(I − λ0 L) ⊕ Range(I − λ0 L)(= X1 ⊕ Y2 ),
(i.e. the algebraic and geometric multiplicities of the characteristic value
of λ0 coincide) and
Q1 Dkx Γ(0) · (x)k , 0 for every xǫX1 − {0}.
Again, from §1 the mapping Dk g(0)(η, x)k verifies the condition (R−
N.D) or not independently of the choice of X2 and Y1 so that we can
take X1 = Y1 in the applications. With this choice, we get the simplified 101
expressions
σηk−1
x + Q1 Γ(σηk−1 , x + ϕ(σηk−1 , x)),
λ0
σk!ηk−1
x + Q1 Dkx Γ(0) · (x)k ǫX1 ,
Dk gσ (0) · (η, x)k = −
λ0
gσ (η, x) = −
(2.37)σ
(2.38)σ
for every (η, x)ǫR × X1 . Also, none of the curves in the local zero set of
gσ has a “vertical” tangent at the origin and each curve in the local zero
set of f is of the from
t → (µ(t), x(t)),
with µ(t) = σηk−1 (t) and t → (η(t), x(t)) is one of the curves in the local
zero set of gσ (thus of class C m−k+1 at the origin and of class C m away
from it). Each non-trivial curve in the local zero set of f is then also of
class C m−k+1 at the origin and of class C m away from it (the argument is
the same as when k is even) and it is tangent to the hyperplane {0} × X1
at the origin. For σ = 1 (respectively - 1), the corresponding curves of
80
3. Applications to Some Nondegenerate problems
the local zero set of f are located in the half space µ ≥ 0 (respectively
µ ≤ 0) because, contrary to what happens when k is even, σηk−1 (t) does
not change sign here.
2
1
3
3
1
2
Figure 2.4: Local zero set of f (k odd).
102
103
The differences with the case when k is even are as follows: (i) Both
mapping (2.38)σ must verify the condition (R − N.D.) for σ = 1 and
σ = −1. Charging η into eiπ(k−1) η, it is easy to see that this is the case
if one of them verifies the condition (C − N.D.), because this property is
invariant under C-linear changes of variables.
(ii) Bifurcation (i.e. existence of notrivial curves) is not ensured (an
example was given in Chapter 1). However, bifurcation occurs if η is
odd (Theorem 1.2 of Chapter 1). This means that for at least one of the
values σ = 1 or σ = −1, the zero set of the mapping Dk gσ (0) · (η, x)k
contains at least one nontrivial line.
Remark 2.5. Theorem 1.2 of Chapter 1 (Krasnoselskii’s theorem) is
based on topological degree arguments. Is there however a purely algebraic proof of the above statement?
(iii) Let νσ , σ = ±1 denote the number of curves in the local zero set
of the mapping gσ . Then, a priori, the number of curves in the local zero
set of f is ν1 + ν−1 . Actually, ν1 and ν−1 must be odd and the number of
distinct curves in the local zero set of f is (ν1 + ν−1 )/2 (hence ≤ kn again
since νσ ≤ kn ). Indeed, it is clear from the evenness of k − 1 that when
a curve (η(t), x(t)) is in the local zero set of gσ , the curve (−η(t), x(t))
3.2. Application to Problems of Bifurction.....
81
is also in it. Both provide the same curve (µ(t), x(t)) = (σηk−1 (t), x(t))
in th local zero set of f and the two vectors ((dη/dt)(0)(dx/dt)(0)) and
(−(dη/dt)(0), (dx/dt)(0)) are not collinear if and only if (dx/dt)(0) , 0,
because (dη/dt)(0) is , 0 as we observed earlier5 . Now the condition
(dx/dt)(0) , 0 is fulfilled by all the curves in the local zero set of gσ , except the trivial branch (recall that the correspondence between the curves
in the local zero set of gσ and their tangents at the origin is one-to-one).
Hence, each non-trivialcurve in the local zero set of f is provided by two
distinct non-trivial curves in the local zero set of g1 or by two distinct
non-trivial curves in the local zero set of g−1 . Thus, ν1 and ν−1 must be
odd and the number of non-trivial curves in the local zero set of f is
ν1 − 1 ν−1 − 1 ν1 + ν−1
+
=
− 1.
2
2
2
Adding the trivial branch, we find (ν1 + ν−1 )|2 to be the number 104
of distinct curves in the local set of f . Exactly (ν1 − 1)|2 of them are
supercritical (i.e. located in the half-space µ ≥ 0 of R × X1 ) and exactly
(ν−1 − 1)|2 are subcritical (i.e. located in the half space µ ≤ 0 of R × X1 ).
One (the trivial branch) is transcritical. Of course, it may happen that
ν1 = 0 or ν−1 = 0 (or both).
Remark 2.6. Recall that in both the case (k even and k odd), we made
the a priori assumption X1 1 Y2 (and it turned out that the conditions
X1 ∩ Y2 = {0} was necessary). If X1 ⊂ Y2 , the trick of changing the
parameter µ into ηk−1 , σ = ±1 does not work: Indeed, the first nonzero
derivative of g or gσ at the origin remains of order k and its value at the
point (η, x)ǫR × X1 repeated k times reduces to
(η, x)ǫR × X1 → Q1 Dkx Γ(0) · (x)k ǫY1 .
This mapping never verifies the condition (R − N.D.) because of the
trivial branch in its zero set at which its derivative vanishes.
Remark 2.7. We leave it to the reader to check that changing the paramete µ into a ηk−1 , a , 0, when k is even is equivalent to changing it into
5
odd.
In comment 2.4 and when k is even, but the argument can be repeated when k is
3. Applications to Some Nondegenerate problems
82
105
ηk−1 (i.e. a = 1) as we have done. No more generality will be reached
either by changing µ into a ηk−1 + o(ηk−1 ), a , 0 and no other exponent
p , k − 1 can be used in the change µ = η p (because the condition
(R − N.D.) always fails with such a choice). Similarly, when k is odd,
changing µ into aηk−l + ◦(ηk−l ), a > 0 is equivalent to changing µ into
ηk−l and changing µ into aηk−1 + o(ηk−1 ) is equivalent to changing µ into
−ηk−1
To complete this section, we come back to the case n = 1 for comparisons and further comments. In Chapter 1, we have solves the probj
lem without any assumption on the derivatives Q1 D x Γ(0)|(X1 ) j and the
condition (R − N.D.) was seen to be equivalent to the fact that the algebraic multiplicity of λ0 equals its geometric multiplicity i.e.
X = Ker(I − λ0 L) ⊕ Range(I − λ0 L)(= X1 ⊕ Y2 ).
(2.39)
Neverthless, if there is an integer 3 ≤ k ≤ m6 such that
Q1 D xj Γ(0) = 0, 0 ≤ j ≤ k − 1,
Q1 Dkx Γ(0)|(X1 )k , 0,
106
(2.40)
the trick of changing the parameter µ into ηk−1 (k even) or σηk−1 , σ = ±1
(k odd) works as when n is ≥ 2 and the analysis we have made in this
section can be repeated with no modification. In particular, condition
(2.39) remains necessary. Here, it is also a sufficient condition for the
mapping Dk g(0) · (η, x)k (2.20) (or Dk gσ (0) · (η, x)k ((2.38)σ )) to verify
the condition (R − N.D.). Indeed, as X1 and Y1 are one-dimensional, we
can write xǫX1 in the form
x = tx0 ,
for some tǫR, where x0 is a given non-zero element of X1 . Also let y0
be a given non-zero element of Y1 7 and consider the linear continuous
form y∗ ǫX ′ (= Y ′ ) characterized by (cf. Chapter 1, §3)



hy∗ , y0 i = 1,


hy∗ , yi = 0 for every yǫY2 .
6
If k = 2, see Comment 2.1.
From (2.39), the choice x0 = y0 is available and brings slight simplifications in the
formulae.
7
3.2. Application to Problems of Bifurction.....
83
This linear form allows us to express the projection operator Q1 onto
the space Y1 . More precisely
Q1 y = hy∗ , yiy0 ,
for every yǫX(= Y). Then, for k even, we find from (2.20) that
#
"
k!ηk−1 ∗ 0
k ∗
k
0 k
k
k
hy , x i + t hy , D x Γ(0) · (x ) i y0
D g(0) · (η, x) = −
λ0
and proving that this mapping verifies the condition (R − N.D.) amounts
to proving that the real-valued mapping of the two real variables η, t
given by
k!ηk−1 ∗ 0
hy , x i + tk hy∗ , Dkx Γ(0) · (x0 )k iǫR,
λ0
does the same, which is always the case because of the relations hy∗ , 107
x0 i , 0 (since X1 ∩ Y2 = {0}) and hy∗ , Dkx Γ(0) · (x0 )k i , 0 (since
Q1 Dkx Γ(0)|(X1 )k , 0). Besides, as expected, the zero set of Dk g(0) · (η, x)k
is the union of the trivial branch and exactly one nontrivial line, namely
that line for which
"
#1/k−1
hy∗ , Dkx Γ(0) · (x0 )k i
η, ηǫR.
t = k!
λ0 hy∗ , x0 i
(η, t)ǫR2 → −
Repeating the arguments we used in the case n ≥ 2, we see that the
corresponding non-trivial branch in the local ero set of is tangent to the
asis {0} × X1 at the origin and located on both sides of it: we are in the
presence of a phenomenon of transcritical bifurcation.
Figure 2.5: Local zero set of f when n = 1, k even ≥ 4
84
3. Applications to Some Nondegenerate problems
Analogously, when k is odd, one has (cf. ((2.38)σ ))
#
"
σηk−1 t ∗ 0
k ∗
k
0 k
k
k
hy , x i + t hy , D x Γ(0) · (x ) i y0 .
D gσ (0) · (η, x) = −k!
λ0
108
If so, because k−1 is even, the local zero set of this mapping reduces
to the trivial branch if
" ∗ k
#
hy , D x Γ(0) · (x0 )k i
sgn
= sgn(−σ)
λ0 hy∗ , x0 i
and is the union of the trivial branch and two nontrivial lines, namely,
those for which
#1/k−1
"
hy∗ , Dkx Γ(0) · (x0 )k i
η, ηǫR,
t = ± k!σ
λ0 hy∗ , x0 i
if
"
#
hy∗ , Dkx Γ(0) · (x0 )k i
sgn
= sgnσ.
λ0 hy∗ , x0 i
They both provide the same curve in the local zero set of f , located
on one side of the axis {0} × X1 . The bifurcation is either supercritical
or subcritical depending on which among the mappings Dk g1 (0) · (η, x)k
and Dk g−1 (0) · (η, x)k possesses the non-trivial lines in its zero set
Figure 2.6:
3.3. Application to a Problem......
109
85
Remark 2.8. Though the assumptions of this section are stronger than
those we made in Chapter 1 when n = 1, the regularity of the non-trivial
curve in the local zero set of f is only found to be of class C m−k+1 at the
origin (but we know it is actually of class C m−1 ). In contrast, the location
of this curve with respect to the axis {0} × X1 could not be derived from
the analysis of Chapter 1 because it clearly depends on the evenness of
k, which was ignored in Chapter 1.
Remark 2.9. The assumption (2.40) can be replaced by
j
D x Γ(0)|(X1 ) j = 0, 0 ≤ j ≤ k − 1,
Q1 Dkx Γ(0)|(X1 )k , 0,
(2.41)
or by a suitable combination (2.40) and (2.41) such as in Remark 2.3.
3.3 Application to a Problem with no Branch of
Solutions Known a Priori.
We shall here consider the problem of finding the “small” solution (µ,
x)ǫR × X an equation of the form
F(x) = µy0
(3.1)
where F is a mapping of class C m , m ≥ 1, in a neighbourhood of the
origin in the Banach space X, with values in another Banch space Y,
such that F(0) = 0 and where y0 is a given element of Y. In order to deal 110
with a situation essentially different from that of §2 we shall assume
y0 < RangeD x F(0).
(3.2)
First, we write the problem in the form
G(µ, x) = 0,
with
G(µ, x) = F(x) − µy0 .
(3.3)
3. Applications to Some Nondegenerate problems
86
Due to (3.2), the space Y2 = RangeDG(0) is
Y2 = RangeDG(0) = Ry0 ⊕ RangeD x F(0)
(3.4)
whereas, setting
X1 = KerD x F(0),
one has
e1 = KerDG(0) = {0} × X1 .
X
111
(3.6)
As required in these notes, we shall assume that DG(0)ǫL (R× X, Y)
is a Fredholm operator with index 1, which will be for instance, the case,
if D x F(0) is a Fredholm operator with index 0. As usual, n will denote
the codimension of the space Y2 .
When n = 0, the same problem has already been encountered in
Chapter 1. The situation was simple because the Implict function theorem applied so that the local zero set of G was found to be made up
of exactly one curve of class C m . Recall that this curve has a “vertical”
tangent (i.e. the one - dimensional space {0} × X1 ) at the origin and we
mentioned that two typical cases where those when the origin is either
a “turning point” or a “hysteresis point” (cf. Fig. 3.2 of Chapter 1). A
detailed explanation of these phenomena will be given later on. For the
time being, we shall assume n ≥ 1. Performing the Lyapunov-Schmidt
reduction as described in Chapter 1, §2 and setting e
x = (µ, x), we get a
reduced equation of the general form
f (e
x) = Q1G(e
x+e
ϕ(e
x)),
e1
where e
x belongs to some neighbourhood of the origin in the space X
and Q1 denotes the projection operator onto some given complement of
e1 identifies itself with
the space Y2 . Because of (3.6), the variable e
xǫ X
e2 of the space
the variable xǫX1 . Provided we choose the complement X
e1 of the form
X
e2 = R × X2 ,
X
where X2 is a given topological complement of X1 , the mapping e
ϕ can
be identified with a pair
e
ϕ(x) = (µ(x), ϕ(x))ǫR × X2 ,
3.3. Application to a Problem......
87
of mappings of class C m aroun the origin.
As Q1 y0 = 0 (cf. (3.4)), the reduced equation takes the form
f (x) = Q1 F(x + ϕ(x)) = 0.
(3.7)
The results of Chapter 2, §4 will be available if the first nonzero 112
derivative of the above mapping at the origin verifies the condition (R −
N.D.). A general frame work in which it takes a very simple expression
is as follows: Consider the integer 2 ≤ k ≤ m characteried by

j


 Q1 D x F(0)|(X1 ) j = 0, 0 ≤ j ≤ k − 1,
(3.8)


 Q1 Dkx F(0)|(X )k , 0,
1
(if such a k exists of course) and let k1 and k1 be defined by
j
k1 = min{0 ≤ j ≤ k, Q1 D x F(0) , 0},
j
k1 = min{0 ≤ j ≤ k, D x F(0)|(X1 ) j , 0},
(3.9)
(3.10)
so that 2 ≤ k1 , k1 ≤ k. Under the condition8
k ≤ k1 + k1 − 2,
(3.11)
the first non-zero derivative of the mapping f in (3.7) at the origin is of
order k and its value at the point xǫX1 repeated k times is
Dk f (0) · (x)k = Q1 Dkx F(0) · (x)k .
(3.12)
The reader can easily check this assertion in the two frequently encounted particular cases k = k1 or k = k1 , namely when
j
Q1 D x F(0) = 0, 0 ≤ j ≤ k − 1,
or
D xj F(0)|(X1 ) j = 0, 0 ≤ j ≤ k − 1.
113
8
Y1 .
Observe that none of the integers k, k1 and k1 depends on the choice of the space
3. Applications to Some Nondegenerate problems
88
For a general result, see [31]. When (3.11) holds and the mapping
(3.12) verifies the condition (R − N.D.) (a property which is easily seen
to be independent of the choice of the space Y1 ) the largest number of
curves in the local zero set of f - hence of G - is kn . Note here that
n = dimX1 − 1(= dimKerD x F(0) − 1).
(3.13)
Of course, the curves are of class C m−k+1 at the origin and of class
C m away from it.
Remark 3.1. In contrast to the case n = 0, it is quite possible that the
origin is an isolated solution. For instance, let X = Y = R3 and choose
F as

 2
 x1 + x22 + x23 


0
F(x1 , x2 , x3 ) = 


x3
while y0 is taken as
 
0
 
0
y = 1 .
 
0
The only solution of the equation F(x1 , x2 , x3 ) = µy0 is µ = x1 =
x2 = x3 = 0. Nevertheless, (3.8) is fulfilled with k = 2 and (3.11) holds.
The mapping (3.13) associated with this example is
(x1 , x2 , x3 )ǫR3 → x21 + x22 + x23 ǫR
114
and verifies the condition (R − N.D.) trivially.
Observe, however, that existence of at least one curve of solution is
generated by comment 1.3 of Chapter 2 when k is odd.
The analysis we have made does not provide any information on
the location of the curves. Incidentally, such information was obtained
in the problem we considered in the previous section after we had to
make a change of parameter to find the structure of its set of solutions.
Here, one has no such motivation indeed since a satisfactory answer has
already been given to this question but it is not without interest to see
3.3. Application to a Problem......
89
where a change of parameter could lead us: setting µ = η p for some
integer p ≥ 2, we have to solve the problem
Ĝ(η, u) = G(η p , x) = F(x) − η p y0 = 0.
(3.14)
The main difference with the case when the parameter µ is unchanged is that (compare with (3.4) and (3.6))
Ŷ2 = RangeDĜ(0) = RangeD x F(0),
ê
X 1 = KerDĜ(0) = R × KerD x F(0) = R × X1 .
(3.15)
(3.16)
Thus, the mapping DĜ(0) will be a Fredholm operator with index
1 if and only if the mapping D x F(0) is a Fredholm operator with index 0 (namely, Range D x F(0) must be closed). Also, with the previous
definition of n (= codim Y2 ), one has
codimŶ2 = n + 1(= n̂) ≥ 1,
dim Ker ê
X = n + 2(= n̂ + 1) ≥ 2.
(3.17)
(3.18)
We shall denote by Ŷ1 any complement of Ŷ2 and call Q̂1 and Q̂2 115
the projection operators onto Ŷ1 and Ŷ2 respectively. Performing the
Lyapunov-Schmidt reduction of the problem (3.14), we find a reduced
equation of the general form
ĝ(ê
x) = Q̂1Ĝ(ê
x + ê
ϕ(ê
x)).
Here, the variable ê
x of the space ê
X 1 = R× X1 is the pair (η, x)ǫR× X1
ê
and the mapping ψ takes its values in some topological complement ê
X2
ê
ê
of the space X 1 . Choosing X 2 = {0} × X2 where X2 is any topological
complement of X1 , the mapping ê
ψ identifies itself with a mapping ψ̂
with values in X2 so that
ĝ(η, x) = Q̂1 F(x + ϕ̂(η, x)) − η p Q̂1 y0
(note that Q̂1 y0 , 0 since y0 < Range D x F(0) by hypothesis). At this
stage, it is necessary for a better understanding to examine how the mapping ϕ̂ is related to the variable µ through the change µ = η p . To this
3. Applications to Some Nondegenerate problems
90
end, write each element xǫX in the form x = x1 + x2 with x1 ǫX1 and
x2 ǫX2 . The equation G(µ, x) = 0 becomes equivalent to the system
Q̂1G(µ, x1 + x2 ) = 0,
Q̂2G(µ, x1 + x2 ) = 0.
As Q̂2 D xG(0) = Q̂2 D x F(0)ǫ Isom (X2 , Ŷ2 ), the second equation is
solved by the Implict function theorem and is equivalent to
x2 = ϕ̂(µ, x1 ),
116
where ϕ̂ is mapping of class C m around the origin in R × X1 with values
in X2 , uniquely determined by the condition ϕ̂(0) = 0. Referring to the
first equation, we find a reduced equation
e
f (µ, x1 ) = Q̂1 G(µ, x1 + ϕ̂(µ, x1 ))
for (µ, x1 ) around the origin in R × X1 . Dropping the index “1” in the
variable x1 ǫX1 and using (3.3), we deduce
fˆ(µ, x) = Q̂1G(µ, x + ϕ̂(µ, x)),
(3.19)
for (µ, x) around the origin of R × X1 . Note that this reduction differs
from the Lyapunov-Schmidt reduction as it was described in Chapter 1
because the space Ŷ2 is not the range of the global derivative DG(0) but
only the range of the partial derivative D xG(0). In particular, Dϕ̂(0) , 0
since Dµ ϕ̂(0) , 0 in general. However, the local zero set of fˆ (3.19)
immediately provides the local zero set of G and a simple verification
shows that the mapping ψ̂ and ϕ̂ are linked through the relation
ψ̂(η, x) = ϕ̂(η p , x).
It follows that
ĝ(η, x) = fˆ(η p , x) = Q̂1 F(x + ϕ̂(η p , x)) − η p Q̂1 y0 .
Again, the problem now is to find the first non-zero derivative of
ĝ at the origin. Of course, it will depend on p but since the condition
3.3. Application to a Problem......
117
91
(R − N.D.) must hold under general hypotheses, it is possible to show
that the only available value of p is p = k̂ where the integer 2 ≤ k̂ ≤ m
is characterized by

j


 Q̂1 D x F(0)|(X1 ) j = 0, 0 ≤ j ≤ k̂ − 1,
(3.20)


 Q̂1 Dk̂x F(0)| k̂ , 0.
(X1 )
The process allowing the selection of p should be described in a
general framework rather than on this particular example but even so,
it remains quite technical and will not be presented here (see [31] for
details).
The mapping ĝ corresponding to the choice p = k̂ is
ĝ(η, x) = fˆ(ηk̂ , x) = Q̂1 F(x + ϕ̂(ηk̂ , x)) − ηk̂ Q̂1 y0 .
(3.21)
Setting
j
k̂1 = min{0 ≤ j ≤ k̂, Q̂1 D x F(0) , 0},
1
k̂ = min{0 ≤ j ≤
j
k̂, D x F(0)|(X1 ) j
, 0},
(3.22)
(3.23)
one has 2 ≤ k̂1 , k̂1 ≤ k̂. Clearly, none of the integers k̂, k̂1 and k̂1 depends
on the choice of the space Ŷ1 and, under the condition
k̂ ≤ k̂1 + k̂1 − 2,
(3.24)
the first non-zero derivative of the mapping ĝ (3.21) at the origin is of
order k̂ with
Dk̂ ĝ(0) · (η, x)k̂ = Q̂1 Dk̂x F(0) · (x)k̂ − k̂!ηk̂ Q̂1 y0 ǫ Ŷ1 ,
(3.25)
for every (η, x)ǫR × X1 . The reader can check there formulae in the
particular cases k̂1 = k̂ or k̂1 = k̂, namely
118
j
Q̂1 D x F(0) = 0, 0 ≤ j ≤ k̂ − 1,
or
j
D x F(0)|(x1 ) j = 0, 0 ≤ j ≤ k̂ − 1.
3. Applications to Some Nondegenerate problems
92
In the general case, see [31]. If the mapping (2.25) verifies the condition (R − N.D.) and k̂ is odd, the local zero set of ĝ provides that of fˆ
(3.19) immediately. If k̂ is even the local zero set of ĝ gives the elements
of the local zero set of fˆ associated with µ ≥ 0. So as to get the elements
associated with µ ≤ 0, the change µ = −ηk̂ is necessary too. When k̂ is
even, we must then examine both the mappings
ĝσ (η, x) = fˆ(σηk̂ , x) = Q̂1 Dk̂x F(0) · (x)k̂ − k̂!σηk̂ Q̂1 y0 ǫ Ŷ1 ,
(3.26)σ
where σ = ±1 and we have
Dk̂ gσ (0) · (η, x)k̂ = Q̂1 Dk̂x F(0) · (x)k̂ − k̂!σηk̂ Q̂1 y0 ,
119
((3.27)σ )
for (η, x)ǫR × X1 .
Assume first that k̂ is odd and the mapping (3.25) verifies the condition (R − N.D.). Arguing as in §1, we see that its zero set contains
no “vertical” line (i.e. no line contained in the hyperplane {0} × X1 ).
Therefore, if t → (η(t), x(t)) is one of the curves in the local zero set of
ĝ, one has
dη
(0) , 0,
dt
so that the curve is located on both sides of the hyperplane {0} × X1 in
R × X1 .
2
1
1
2
Figure 3.1: Local zero set of ĝ (k̂ odd).
As k̂ is odd and the mapping n → ηk̂ is a homeomorphism of R, the
corresponding curve (µ(t), x(t)) with µ(t) = ηk̂ (t) in the local zero set of
3.3. Application to a Problem......
93
fˆ is also located on both sides of the hyperplane {0} × X1 in R × X1 . Of
course,
dx
d
(µ(t), x(t))|t=0 = (0, (0)).
dt
dt
dx
dx
But dt (0) , 0, because the non-zero vector, dη
dt (0), dt (0) is in the
zero set of the mapping (3.25), which does not contain the line (η, 0)9
As a result, each curve (µ(t), x(t)) in the local zero set of fˆ is tangent to
the hyperplane {0} × X1 at the origin. Replacing the hyperplane {0} × X1
in R × X1 by the hyperplane {0} × X in R × X, these statements remain 120
readily valid as concerns the local zero set of the mapping G.
2
1
1
2
Figure 3.2: Local zero set of fˆ (k̂ odd).
Assume next that k̂ is even and the both mappings ((3.26)σ ) verify
the condition (R − N.D.). As before, if t → (η(t), x(t)) is one of the
curves in the local zero set of ĝ1 or ĝ−1 , one has
dη
dx
(0) , 0, (0) , 0.
dt
dt
In particular, the corresponding curve (µ(t), x(t)) with µ(t) = σηk̂ (t)
in the local zero set of fˆ is tangent to the hyperplane {0} × X1 at the
origin. But, as σηk̂ (t) does not change sign for t varying around 0, this
curve is located on one side of the hyperplane {0} × X1 in R × X1 . Again,
replacing the hyperplane {0} × X1 in R × X1 by the hyperplane {0} × X in
R × X, these statements remain readily valid as concerns the local zero 121
9
See Proposition 3.1 later
94
3. Applications to Some Nondegenerate problems
set of the mapping G.
Figure 3.3: Local zero set of fˆ (k̂ even)
We are now going to relate the assumptions we make for solving the
problem after changing the parameter µ into ηk̂ or σηk̂ to the assumptions we made earlier to solve it without changing µ. First, observe that
the integer k̂ characterized by (3.18). Indeed, k̂ is independent of the
choice of the space Ŷ1 and from the definitions, it is immediate that Ŷ1
can be chosen so that
Ŷ1 ⊃ Y1
(3.28)
122
where Y1 is any space chosen for defining k. By the same arguments,
note that
k̂1 ≤ k1 , k̂1 ≤ k1 ,
(3.29)
(cf. (3.9)-(3.10) and (3.22)-(3.23)). But we shall see that the integers
k and k̂ must coincide for the mappings (3.25) or (3.27)σ to verify the
condition (R − N.D.) when n ≥ 1. This is proved in
Proposition 3.1. Assume k̂ is odd (resp. even). Then, the mapping
(3.25) (resp. (3.27)1 and (3.27)−1 ) erifies the condition (R − N.D.) is
and only if
Q̂1 Dk̂x F(0) · (x)k̂ , 0,
(3.30)
for every xǫX1 − {0} and the mapping
xǫX1 → Q1 Dk̂x F(0) · (x)k̂ ǫY1 ,
(3.31)
3.3. Application to a Problem......
95
verifies the condition (R − N.D.). In particular, when n ≥ 1, one must
have k̂ = k. When n = 0, the condition (3.30) holds by the definition of
k̂, k is not defined and Y1 = {0} so that the mapping (3.31) verifies the
condition (R − N.D.) trivially.
Proof. We prove the equivalence when k̂ is odd. When k̂ is even, the
proof is identical and left to the reader.
The condition (3.30) is necessary. Indeed, if Q̂1 Dk̂x F(0) · (x)k̂ = 0
for some xǫX1 − {0}, the line {0} × Rx is in the zero set of the mapping
(3.25) whose derivaitve at (0, x)ǫR × X1 is
(η′ , x′ )ǫR × X1 → k̂ Q̂1 Dk̂x F(0) · ((x)k̂−1 , x′ )ǫ Ŷ1 .
Its null-space contains the two-dimensional subspace R(1, 0)⊕({0}× 123
Rx) and hence its range is of dimension ≤ n̂ − 1 so that the condition
(R − N.D.) fails.
Now, we prove that the mapping (3.31) verifies the condition (R −
N.D.). As our assumptions are independent of the space Ŷ1 , we can
suppose that
Ŷ1 = Ry0 ⊕ Y1 .
(3.32)
If so, Q̂1 y0 = y0 , Q1 Q̂1 = Q1 and the operator Q̂1 − Q1 is the
projection onto the space Ry0 associated with the decomposition Y =
Ry0 ⊕ (Y1 ⊕ RangeD x F(0)). Then, the mapping (3.25) becomes
(η, x)ǫR × X1 → k̂!ηk̂ y0 + Q̂1 Dk̂x F(0) · (x)k̂ ǫ Ŷ1 .
Let now x be a non-zero element of the zero set of the mapping
(3.31) so that Q̂1 Dk̂x F(0) · (x)k̂ = (Q̂1 − Q1 )Dk̂x F(0) · (x)k̂ , and, from the
above, the right hand side is collinear with y0 . Therefore
Q̂1 Dk̂x F(0) · (x)k̂ = λy0 ,
for some real number λ. As k̂ is odd, there is ηǫR such that k̂!ηk̂ =
−λ and the pair (η, x)ǫR × X1 is in the zero set of the mapping (3.25).
Let yǫY1 be given. In particular, yǫ Ŷ1 (cf. 3.32) and there is a pair
(η′ , x′ )ǫR × X1 such that
k̂(k̂!)ηk̂−1 η′ y0 + k̂ Q̂1 Dk̂x F(0) · ((x)k̂−1 , x′ ) = y.
3. Applications to Some Nondegenerate problems
96
Taking the projection onto Y1 , we find
k̂Q1 Dk̂x F(0) • ((x)k̂−1 , x′ ) = y,
124
which proves our assertion.
Conversely, let (η, x)ǫR × X1 be a non-zero element of the zero set of
the mapping (3.25). Observe that η , 0 from (3.30) and it is obvious that
x , 0. Taking the projection onto Y1 , we find that x is in the zero set of
the mapping (3.31). Let ŷǫ Ŷ1 be given and set y = Q1 ŷ. By hypothesis,
there exists x′ ǫX1 such that k̂Q1 Dk̂x F(0) · ((x)k̂−1 , x′ ) = y. As (Q̂1 − Q1 )
is the projection onto the space Ry0 and by definition of y, we find
k̂ Q̂1 Dk̂x F(0) · ((x)k̂−1 , x′ ) = ŷ + λy0 ,
for some real number λ. Setting
η′ = −λ/k̂(k̂!)ηk̂−1 ,
it is clear that
k̂(k̂!)k̂−1 η′ y0 + k̂ Q̂1 Dk̂x F(0) · ((x)k̂−1 , x′ ) = ŷ,
so that the mapping (3.25) verifies the condition (R − N.D.).
The end of our assertion is obvious. If n ≥ 1, we already know that
k̂ ≤ k and, if k̂ < k, one has
Q1 Dk̂x F(0)|(X1 )k̂ = 0,
125
by definition of k, which contradicts the fact that the mapping (3.31)
verifies the condition (R − N.D.). When n = 0, the space Ŷ1 is onedimensional (i.e. n̂ = 1) and the condition (3.30) is then automatically
satisfied by definition of k̂.
To sum up, the local zero set of the mapping G (3.3) can be found
after changing the parameter µ into ηk̂ under the conditions
a) D x F(0) is a Fredholm operator with index 0,
b) k̂ ≤ k̂1 + k̂1 − 2,
3.3. Application to a Problem......
97
c) Q̂1 Dk̂x F(0) · (x)k̂ , 0 for every xǫX1 − {0},
d) the mapping
xǫX1 → Q1 Dk̂x F(0) · (x)k̂ ǫY1 ,
verifies the condition (R − N.D.).
In any case, these assumptions are stronger than those made when
the local zero set of G is determined without chaging the parameter µ
(see in particular (3.29) and recall that k̂ must equal k when n ≥ 1) and
the conclusions are not as precise: The number of curves is found to be
≤ k̂n̂ = k̂n+1 . When n ≥ 1, this upper bound is nothing but kn+1 (but we
know that it is actually kn ) and k̂ when n = 0 (but we know that there
is exactly one curve). The regularity we obtain is of class C m−k̂+1 at
the origin and of class C m away from it, which agress with the previous
result if n ≥ 1 (since k̂ = k) but is not as good when n = 0 (the curve is
actually of class C m at the origin). However, the additional information
which may be of interest is the location of the curves in the space R × X.
In particular, when n = 0, the reader can immediately deduce, from the 126
appropriate discussion given before, that the origin is a “turning point”
when k̂ is even and a “hysteresis point” when k̂ is odd. When n ≥ 1,
bifurcation may occur but the origin can be an isolated solutoion as well
(see the example given in Remark 3.1).
Chapter 4
An Algorithm for the
Computation of the Branches
IN THIS CHAPTER, we describe an algorithm for finding the local zero 127
set of a mapping f verifying the assumptions of Chapter 2. When f is
explicitly known, we obtain an iterative scheme with optimal rate of convergence. However, in the applications to one paramenter problems in
Banach spaces (such as those described in Chapter 3 for instance) f
is the reduced mapping which is known only theoretically, because the
mapping e
ϕ coming from the Lyapunov-Schmidt reduction (cf. Chapter 1, §2) involved in the definition of f is found through the implict
function theorem.
What is explicitly available in this case is only a sequence ( fℓ ) of
mappings tending to f in some appropriate way and the algorithm must
be modified accordingly. Again, optimal rate of convergence can be
established but the proof is more delicate. For the sake of brevity, we
shall only outline the technical differences which occur. The exposition
follows Rabier-E1-Hajji [33]. Full details can be found in Appendix 2.
This chapter is completed by an explicit description of the algorithm in
the case of problems of bifurcation from the trivial branch. A comparison with the classical “Lyapunov-Schmidt method” is given as a conclusion.
99
100
4. An Algorithm for the Computation of the Branches
4.1 A Short Review of the Method of Chapter II.
128
In Chapter 2, we considered the problem of finding the local zero set
of a mapping f of class C m , m ≥ 1, from Rn+1 into Rn (recall that it
is not restrictive to assume that f is defined everywhere) satisfying the
following condition: There is an integer 1 ≤ k ≤ m such that
D j f (0) = 0 0 ≤ j ≤ k − 1,
and the mapping
e
ξǫRn+1 → q(e
ξ) = Dk f (0) · (e
ξ)k ǫRn ,
verifies the condition (R − N.D.). Under this assumption, the set
{e
ξǫS n ; q(e
ξ) = 0}
j
is finite and consists of the 2ν elements e
ξ0 , 1 ≤ j ≤ 2ν. Of course, as
the origin is an isolated solutioon of the equation f (e
x) = 0 when ν = 0,
it is not restrictive to limit ourselves to the case ν ≥ 1 for defining the
algorithm. This assumption will be implicitly made through-out this
chapter.
Setting, for (t, e
ξ)ǫR × Rn+1
k! e
f (tξ) if t , 0,
tk
g(0, e
ξ) = q(e
ξ),
g(t, e
ξ) =
129
it was observed for r > 0 that finding the solution e
x of the equation
f (e
x) = 0 with 0 < ||e
x|| = t < r was equivalent to finding e
x of the form
e
e
x = tξ where
0 < |t| < r, e
ξǫS n ,
g(t, e
ξ) = 0.
Next, an essential step consisted in proving for r > 0 small enough
that the above equation was equivalent to solving the problem
0 < |t| < r, e
ξǫ
2ν
[
j=1
σ j,
4.2. Equivalence of Each Equation with a...
101
g(t, e
ξ) = 0,
where, for each 1 ≤ j ≤ 2ν, σ j denotes an arbitrary neighbourhood of
j
e
ξ0 in S n . Taking the σ j ’s disjoint, the problem reduces to the study of
2ν independent equations
0 < |t| < r, e
ξǫσ j ,
g(t, e
ξ) = 0,
for 1 ≤ j ≤ 2ν where actually, ν of them (properly selected) are sufficient, to provide all the other ν solutions because of symmetry properties. From a practical point of view, it remains to solve the equation
j
g(t, e
ξ) = 0 for |t| small enough and e
ξǫS n around one of the points e
ξ0 .
4.2 Equivalence of Each Equation with a Fixed
Point Problem.
Since the specific value of the index j is not important, we shall denote
j
any one of the points e
ξ0 by e
ξ0 . Given δ > 0, we call △ the closed ball 130
n+1
e
in R
with centre ξ0 and radius δ/2. The diameter δ of the ball △ is
always suppposed to satisfy the condition
0 < δ < 1.
(2.1)
Finally, we denote by C the spherical cap centered at e
ξ0 (playing the
role of the negihbourhood σ j ) defined by
C = △ ∩ Sn
(2.2)
102
4. An Algorithm for the Computation of the Branches
Figure 2.1:
We now establish two simple but crucial geometric properties. As
in Chapter 2, || · || denotes the euclidean norm.
Lemma 2.1. (i) For every e
ξǫ△, every e
ζǫC and every e
τǫTeζ S n , one has
131
|e
ξ −e
τ| ≥ 1 − δ > 0.
(ii) For every e
ξǫC and every e
ζǫC, one has
Rn+1 = Re
ξ ⊕ Teζ S n .
Proof. Recall first, for e
ζǫS n , that
Teζ S n = {e
ζ}⊥ .
To prove (i), we begin by writing
||e
ξ −e
τ|| ≥ ||e
ζ −e
τ|| − ||e
ξ −e
ζ||
Now, since e
ζ and e
τ are orthogonal and e
ζǫC ⊂ S n , we have
Thus, as δ < 1
||e
ζ −e
τ||2 = ||e
ζ||2 + ||e
τ||2 = 1 + ||e
τ||2 .
||e
ξ −e
τ|| > 1 − ||e
ξ −e
ζ|| ≥ 1 − δ > 0.
4.2. Equivalence of Each Equation with a...
103
Next, to prove (ii), it suffices to show that Re
ξ ∩ Teζ S n = {0}, namely
⊥
e
e
that ξ < Teζ S n = {ζ} . But a simple calulation shows that e
ξǫC and e
ζǫC
are never orthogonal where
q
√
δ<2 2− 2
and hence when δ < 1.
Since the mapping q satisfies the condition (R − N.D.) and e
ξ0 is one
j
e
of the points ξ0 , we know that
Dq(e
ξ0 )|Teξ S n ǫI som(Teξ0 S n , Rn ).
0
Lemma 2.2. After shrinking δ > 0 if necessary, one has
Dq(e
ξ)|Teξ S n ǫI som(Teξ S n , Rn ),
for every e
ξǫC. Besides, setting
one has
A(e
ξ) = (Dq(e
ξ)|Teξ S n )−1 ǫL (Rn , TeξS n ) ⊂ L (Rn , Rn+1 ),
132
(2.3)
AǫC 0 (C, L (Rn , Rn+1 )).
Proof. Let B1 be the open unit ball in Rn
B1 = {e
ξ ′ ǫRn ; ||e
ξ ′ || < 1}.
By identifying Rn+1 with the product Rn × Re
ξ0 , the spherical cap C
n
is homeomorphic to the closed ball Bd/2 ⊂ R centred at the origin with
diameter
!1
δ2 2
d = δ 1−
< 1.
16
This homeomorophism is induced by a C ∞ -mapping
1
e
ξ ′ ǫB1 → θ(e
ξ ′ ) = (e
ξ ′ , (1 − ||e
ξ ′ ||2 ) 2 e
ξ0 )ǫRn+1
4. An Algorithm for the Computation of the Branches
104
It is immediately checked that
133
Dθ(e
ξ ′ )ǫI som(Rn , T θ(eξ′ ) S n ) ⊂ L (Rn , Rn+1 ),
(2.4)
D(q • θ)(e
ξ ′ ) = Dq(θ(e
ξ ′ ))|T θ (eξ′ )S n · Dθ(e
ξ ′ ).
(2.5)
for every e
ξ ′ ǫB1 .
The mapping q • θ is of class C ∞ from B1 into Rn and by differentiating
D(q • θ)(e
ξ ′ ) = Dq(θ(e
ξ ′ )) · Dθ(e
ξ ′ ).
Since the mapping Dθ(e
ξ ′ ) takes its values in the space T θ(eξ′ ) S n , this
can be rewritten as
In particular, for e
ξ′ = 0
D(q • θ)(0) = Dq(e
ξ0 )|Teξ S n · Dθ(0).
0
As Dq(e
ξ0 )|Teξ S n ǫI som(Teξ0 S n , Rn ) and Dθ(0)ǫI som(Rn , Teξ0 S n ), we
0
deduce
D(q • θ)(0)ǫI som(R)n .
By continuity, we may assume that d > 0 (or, equivalently, δ > 0)
has been chosen small enough for D(q • θ)(e
ξ ′ ) to be an isomorphism for
′
every e
ξ ǫBd/2 . Together with (2.4), relation (2.5) shows that
Dq(θ(e
ξ ′ ))|T θ(eξ′ )S n = D(q • θ)(e
ξ ′ ) · [Dθ(e
ξ ′ )]−1 ǫI som(T θ(eξ′ ) S n , Rn ). (2.6)
Hence
Dq(e
ξ)|Teξ S n ǫI som(Teξ S n , Rn ),
for every e
ξǫC since θ is a bijection from Bd/2 to C. Now, taking the
inverse in (2.6) and according to the definition of A in (2.3), we get
A(θ(e
ξ ′ )) = [Dq(θ(e
ξ ′ ))|T θ(eξ′ ) S n ]−1 = Dθ(e
ξ ′ ) · (D(q • θ)(e
ξ ′ ))−1 .
134
The continuity of the mappings Dθ (with values in L (Rn , Rn+1 )) and
D(q • θ) (with values in Isom (Rn )) and the continuity of the mapping
L → L−1 in teh set Isom (Rn ) shows that the mapping A •θ is continuous
on the ball Bd/2 . The continuity of A follows since θ is a homeomorphisn
between Bd/2 and C.
4.2. Equivalence of Each Equation with a...
105
Remark 2.1. With an obvious modification of the above arguments, we
see that the mapping A • θ is actually C ∞ around the origin of Rn . As
θ−1 is a chart of the sphere S n centered at the point e
ξ0 , this means that
∞
e
A is of class C on a neighbourhood of ξ0 in the sphere S n . More
generally, replacing the mapping q(e
ξ) by q(t, e
ξ) a similar proof shows
that the mapping
(t, e
ξ) → [Deξ g(t, e
ξ)|Teξ S n ]−1
is of class C ℓ in a neighbourhood of (0, e
ξ0 ) in R × S n whenever Deξ g is of
ℓ
class C . This result was used in the proof of Theorem 4.1 of Chapter 2
(with ℓ = m − k).
Let us now fix 0 < δ < 1 such that Lemma 2.2 holds. Given a triple
e
(t, ζ0 , e
ξ)ǫR × C × △, the mapping
M(t, e
ζ0 , e
ξ) = e
ξ − A(e
ζ0 ) · g(t, e
ξ)ǫRn+1 ,
(2.7)
M(0, e
ζ0 , e
ξ0 ) = e
ξ0 .
(2.8)
is well-defined ane we have (recalling that g(0, e
ξ0 ) = q(e
ξ0 ) = 0)
Taking e
ζ =e
ζ0 and e
τ = A(e
ζ0 ) · g(t, ξ) in Lemma 2.1(i), we get
Thus, the mapping
||M(t, e
ζ0 , e
ξ)|| ≥ 1 − δ > 0.
M(t, e
ζ0 , e
ξ)
ǫS n ,
N(t, e
ζ0 , e
ξ) =
||M(t, e
ζ0 , e
ξ)||
135
(2.9)
is well-defined in R × C × △ and
N(0, e
ζ0 , e
ξ0 ) = e
ξ0 .
Theorem 2.1. Let (t, e
ζ0 )ǫR × C be given.
(2.10)
(i) Let e
ξǫC be such that g(t, e
ξ) = 0. Then e
ξ is a fixed point of the
mapping N(t, e
ζ0 , ·)
106
4. An Algorithm for the Computation of the Branches
(ii) Conversely, let e
ξǫ△ be a fixed point of the mapping N(t, e
ζ0 , ·).
e
Then, ξǫC and g(t, e
ξ) = 0.
Proof. Part (i) is obvious since, for e
ξǫC such that g(t, e
ξ) = 0 one has
N(t, e
ζ0 , e
ξ) = e
ξ/||e
ξ|| = e
ξ. Conversely, the mapping N(t, e
ζ0 , ·) takes its
e
values in S n so that any fixed point ξǫ△ of the mapping N(t, e
ζ0 , ·) must
belong to △ ∩ S n = C. Next, the relation N(t, e
ζ0 , e
ξ) = e
ξ also becomes
e
ξ(1 − ||e
ξ − A(e
ζ0 ) · g(t, e
ξ)||) = A(e
ξ0 ) · g(t, e
ξ).
But the left hand side is collinear with e
ξ, while the right hand side
belongs to the space Teζ0 S n . By lemma 2.1(ii), both sides must vanish.
In prticular, A(ζe0 ) · g(t, e
ξ) = 0 and hence g(t, e
ξ) = 0 since the linear
operator A(e
ζ0 ) is one-to-one.
136
Remark 2.2. As announced, the problem reduces to finding the fixed
points of a given mapping. We emphasize that it is essential, in practice,
that the vector e
ζ0 be allowed to be arbitrary in C. Indeed, the “natural”
choice e
ζ0 = e
ξ0 is not realistic in the applications beacuse e
ξ0 is not known
exactly in general.
4.3 Convergence of the Successive Approximation
Method.
From Theorem 2.1, a constructive method for finding the solutions of
the equation g(t, e
ξ) = 0 in R × C consists in applying the successive
approximation method to the mapping
e
ξǫ△ → N(t, e
ζ0 , e
ξ)ǫC,
(3.1)
where e
ζ0 ǫC is arbitrarily fixed. In what follows, we prove after shrinking δ if necessary that there is r > 0 such that the mapping (3.1) is a
contraction of △ into itself for every pair (t, e
ζ)ǫ(−r, r) × C.
As a preliminary step, let us observe, by the continuity of the mapping A (Lemma 2.2) and by the continuity of g and Deξ g on R × Rn+1 (cf.
Chapter 2, Lemma 3.3), that the mapping M is continuous on R × C × △
4.3. Convergence of the Successive Approximation Method.
107
and its partial derivative Deξ M exists and is continuous on R × C × △.
More precisely, for every e
hǫRn+1 ,
Deξ M(t, e
ζ0 , e
ξ) · e
h =e
h − A(e
ζ0 ) · Deξ g(t, e
ξ) · e
h.
In particular, with (t, e
ζ0 , e
ξ) = (0, e
ξ0 , e
ξ0 ) we find:
Deξ M(t, e
ξ0 , e
ξ0 ) · e
h =e
h − A(e
ξ0 ) · Deξ g(0, e
ξ0 ) · e
h.
(3.2)
But Deξ g(0, e
ξ0 ) = Dg(e
ξ0 ) and by definition of A, it follows that
Deξ M(0, e
ξ0 , e
ξ0 )|Teξ S n = 0.
0
137
Next, taking e
h=e
ξ0 in (3.2) we find
Deξ M(0, e
ξ0 ; e
ξ0 ) · e
ξ0 = e
ξ0 − A(e
ξ0 ) · Deξ q(0, e
ξ0 ) · e
ξ0 = e
ξ0 ,
because Deξ q(0, e
ξ0 ) = kq(e
ξ0 ) = 0. The space Rn+1 being the orthogonal
direct sum of the spaces Re
ξ0 and Te S n , these relations mean that the
ξ0
ξ0 , e
ξ0 ) is nothing but the orthogonal projection
partial dericative Deξ M(0, e
onto the space Re
ξ0 , namely
ξ0 , e
ξ0 ) · e
h = (e
ξ0 |e
h)e
ξ0 ,
Deξ M(0, e
for every e
hǫRn+1 where (·|·) denotes the canonical inner product of Rn+1 .
As a result, the mapping N is continuous on R × C × △ and its partial
derivative Deξ N exists and is continuous on R × C × △. We now proceed
to show that
ξ0 , e
ξ0 ) = 0.
Deξ N(0, e
An elementary computation provides
Deξ N(t, e
ζ0 , e
ξ) · e
h=
Deξ M(t, e
ζ0 , e
ξ) · e
h
||M(t, e
ζ0 , e
ξ)||
−
(Deξ M(t, e
ζ0 , e
ξ) · e
h|M(t, e
ζ0 , e
ξ))
||M(t, e
ζ0 , e
ξ)||3
M(t, e
ζ0 , e
ξ).
With (t, e
ζ0 , e
ξ) = (0, e
ξ0 , e
ξ0 ) and since M(0, e
ξ0 , e
ξ0 ) = e
ξ0 and Deξ M(0,
e
e
e
e
e
ξ0 , ξ0 ) · h = (ξ0 |h), we get
Deξ N(t, e
ζ0 , e
ξ) · e
h = (e
ξ0 |e
h)e
ξ0 − ((e
ξ0 |e
h)e
ξ0 |e
ξ0 ) = 0.
108
4. An Algorithm for the Computation of the Branches
Theorem 3.1. Let 0 < γ < 1 be a given constant. After shrinking 138
δ > 0 if necessary, there exists r > 0 such that the mapping N(t, e
ζ, ·) is
Lipschitz-continuous with constant γ from the ball △ into itself for every
pair (t, e
ζ)ǫ[−r, r] × C.
Proof. From the relation Deξ N(0, e
ξ0 , e
ξ0 ) = 0 and the continuity of Deξ N
in R × C × △, we deduce that there is a neighbourhood of (0, e
ξ0 , e
ξ0 )
n+1
in R × S n × R
in which ||Deξ N(t, e
ζ0 , e
ξ)|| is bounded by γ. Clearly,
after shrinking δ (hence △ and C simultaneously), the neighbourhood
in question can be taken as the product [−r, r] × C × △ for r > 0 small
enough. Hence, by Taylor’s formula (since △ is convex)
||N(t, e
ζ0 , e
ξ) − N(t, e
ζ0 , e
ζ)|| < γ||e
ξ −e
ζ||
for every (t, e
ζ0 )ǫ[−r, r] × C and every e
ξ, e
ζǫ△. In other words, for every
e
e
(t, ζ0 )ǫ[−r, r] × C, the mapping N(t, ζ, ·) is Lipschitz-continuous with
constant γ in the ball △. The property is obviously not affected by arbitrarily shrinking r > 0. Let us then fix δ as above, which determines the
ball △ and the spherical cap C. We shall complete the proof by showing, for r > 0 small enough, that the mapping N(t, e
ξ0 , ·) maps the ball △
e
into itself for every point (t, ζ0 )ǫ[−r, r] × C. Indeed, for e
ξǫ△ and since
N(0, e
ζ0 , e
ξ0 ) = e
ξ0 for every e
ζ0 ǫC (cf. (2.10)), one has
139
||N(t, e
ζ0 , e
ξ) − e
ξ0 || = ||N(t, e
ζ0 , e
ξ) − N(0, e
ζ0 , e
ξ0 )|| ≤
≤ ||N(t, e
ζ0 , e
ξ) − N(t, e
ζ0 , e
ξ0 )|| + ||N(t, e
ζ0 , e
ξ0 ) − N(0, e
ζ0 , e
ξ0 )||.
The first term is bounded by γ||e
ξ −e
ξ0 ||, thus by γδ/2. Finally, due
to the uniform continuity of the function N(·, ·, e
ξ0 ) on compact sets, we
deduce that r > 0 can be chosen so that the second term is bounded by
(1 − γ)δ/2 and our assertion follows.
4.4 Description of the Algorithm.
The algorithm of computation of the solutions of the equation g(t, e
ξ) =
0 in [−r, r] × C is immediate from Theorem 3.1 : fixing tǫ[−r, r] and
4.4. Description of the Algorithm.
109
choosing an arbitrary element e
ξ0 ǫC, we define the sequence (e
ζℓ ) through
the formula
e
ζℓ+1 = N(t, e
ζ0 , e
ζℓ ), ℓ ≥ 0.
(4.1)
This sequence tends to the unique fixed point e
ξǫ△ of the mapping
e
N(t, ζ0 , ·) which actually belongs to C and is a solution of the equation
g(t, e
ξ) = 0 (Theorem 2.1).
For t , 0, the sequence
e
xℓ = te
ζℓ
tends to e
x = te
ξ, one among the 2ν solutions with euclidean norm |t| of
the equation f (e
x) = 0. Of course, for t = 0, the sequence (e
ζℓ ) tends to
e
ξ0 , unique solution in C of the equation g(0, e
ξ)(= q(e
ξ)) = 0.
Remark 4.1. Incidentally, observe that the above algorithm allows to
find arbitrarily close approximations of e
ξ0 when only a “rough” estimate
e
(i. e. ζ0 ) of it is known.
Since our algorithm uses nothing but the contraction mapping prin- 140
ciple, the rate of convergence is geometrical. More precisely
||e
ζℓ − e
ξ|| < γℓ ||e
ζ0 − e
ξ||,
for every ℓ ≥ 0, where 0 < γ < 1 is the constant involved in Theorem
3.1. Thus as the points e
ζ0 and e
ξ belong to the ball △, one has ||e
ζ0 − e
ξ|| <
δ < 1 and hence
||e
ζℓ − e
ξ|| ≤ γℓ δ ≤ γℓ ,
for ℓ > 0. Multiplying these inequalities by |t|, we get
||e
xℓ − e
x|| ≤ γℓ ||e
x0 − e
x|| ≤ |t|γℓ ,
for ℓ ≥ 0.
Let us now make the scheme (4.1) explicit. As
g(t, e
ξ) =
k! e
f (tξ) for t , 0,
tk
(4.2)
110
4. An Algorithm for the Computation of the Branches
and by the definition of the mapping N in (2.9), we get









e
e
ζℓ+1 = eζℓ ,
||ζℓ ||
e
ζℓ = e
ζℓ − k!
A(e
ζ0 ) · f (te
ζℓ ), ℓ ≥ 0.
tk
(4.3)
A scheme for the computation of the sequence (e
xℓ ) can be immediately derived. Since e
x0 = te
ζ, the iterate e
xℓ+1 = te
ζℓ+1 is defined by
141



xℓ+1 = |t| ||eexxℓℓ || ,
 e


k!
 e
A(e
ζ0 ) · f (e
xℓ ), ℓ ≥ 0.
xℓ = e
xℓ − tk−1
(4.4)
Remark 4.2. When t = 0, the sequence (e
ζℓ ) is defined by









e
e
ζℓ+1 = eζℓ ,
||ζℓ ||
e
ζℓ = e
ζℓ − A(e
ζ0 ) · q(e
ζℓ ), ℓ ≥ 0.
Of course, the sequence (e
xℓ ) is the constant one e
xℓ = 0.
Remark 4.3. (Practical method): Note that the computation of the term
A(e
ζ0 ) · f (e
xℓ ),
does not require the explicit form of the linear operator A(e
ζ0 ). Indeed,
by means of classical algorithm for solving linear systems, it is obtained
as the solution e
τǫTeζ0 S n = {e
ζ0 }⊥ of the equation
Dq(e
ζ0 ) · e
τ = f (e
xℓ ).
(4.5)
From the fact that the components of the mapping q are polynomials and hence are completely determined by a finite number of coefficients (depending on the derivative Dk f (0)), the linear mapping Dq(e
ζ0 )
is completely determined by e
ζ0 and these coefficients. The tangent space
ζ0 }⊥ also is completely determined by e
ζ0 , which makes it posTeζ S n = {e
sible to solve the equation (4.5) in practice.
4.5. A Generalization to the Case.....
142
111
Remark 4.4. The case k = 1 is the case when the Implicit function theorem applies. The mapping q is the linear mapping D f (0) and the vector
e
ξ0 is a normalized vector of the one-dimensional space Ker D f (0). The
sequence (e
xℓ ) is defined by the simples relation (after choosing e
x0 = te
ζ0 )



xℓ+1 = |t| ||eexxℓℓ || ,
 e


 e
xℓ = e
xℓ − (D f (0)|{eζ0 }⊥ )−1 f (e
xℓ ), ℓ ≥ 0,
where e
ζ0 ǫS n is “close enough” to e
ξ0 . This sequence tends to one of the
(two) solutions with eucliden norm |t| of the equation f (e
x) = 0.
4.5 A Generalization to the Case When the Mapping f is not Explicitly Known.
For the reasons we explained at the beginning of this chapter, it is important to extend the algorithm described in §4 to the case when the
mapping f is not explicitly known but a sequence ( fℓ ) tending to f is
available.
More precisely, the mapping f verifying the same assumptions as
before, we shall assume that there is an open neighbourhood O of the
origin in Rn+1 such that f is the limit of ( fℓ ) in the space C k (O, Rn )
(equipped with its usual Banach space structure). Besides, we shall assume
D j fℓ (0) = 0, 0 ≤ j ≤ k − 1,
(5.1)
D fℓ (0) = D f (0),
(5.2)
k
k
so that the first k derivatives of f and fℓ at the origin coincide for every
ℓ ≥ 0.
Remark 5.1. The conditions (5.1)-(5.2) may seem very restrictive. 143
However, they are quite adapted to the applications to one-parameter
problems in Banch spaces, as we shall see later on.
If this is the case, the mapping
e
ξǫRn+1 → q(e
ξ) = Dk f (0) · (e
ξ)k ǫRn ,
(5.3)
112
4. An Algorithm for the Computation of the Branches
can as well be defined through any of the mappings fℓ , namely
q(e
ξ) = Dk fℓ (0)(e
ξ)k ,
(5.4)
for every ℓ ≥ 0. As usual, we shall assume that it verifies the condition
(R − N.D.).
In analogy with §§1 and 2. let us define, for every ℓ ≥ 0,
k! e
fℓ (tξ) for t , 0,
tk
gℓ (0, e
ξ) = q(e
ξ),
Mℓ (t, e
ζ0 , e
ξ) = e
ξ − A(e
ζ0 ) · gℓ (t, e
ξ),
gℓ (t, e
ξ) =
Mℓ (t, e
ζ0 , e
ξ)
Nℓ (t, e
ζ0 , e
ξ) =
.
||Mℓ (t, e
ζ0 , e
ξ)||
144
(5.5)
(5.6)
(5.7)
(5.8)
Of course, the key point is that a uniform choice of 0 < δ < 1 can be
made for defining the mapping Mℓ and Nℓ in (5.7)-(5.8) because the way
of shrinking δ in §2 before Theorem 2.1 depends only upon geometric
properties of the sphere S n and upon the mapping q (Lemma 2.2) which
is equivalently defined through f or any of the fℓ ’s.
The natural question we shall answer is to know whether 0 < δ < 1
and r > 0 can be found so that the sequence



ζ0 ǫC,
 e
(5.9)


 e
ζℓ+1 = Nℓ (t, e
ζ0 , e
ζℓ ), ℓ ≥ 0,
is well defined and tends to a (unique) fixed point of the mapping N(t,
e
ζ0 , ·). In such a case, the algorithm problem is solved through Theorem
2.1 and the review given in §1.
We shall only sketch the proof of the convergence of the scheme
(5.9), in which the main ideas are the same as before. The interested
reader can refer to Appendix 2 for a detailed exposition.
The first step consists in proving, for any r > 0 and δ > 0 small
enough, the the sequence (gℓ ) (respectively Deξ gℓ ) tends to g (respectively Deξ g) in the space C 0 ([−r, r] × △, Rn )(resp.C 0 [−r, −r] × C × ∆,
Z (Rn+1 , Rn )). This follows from the relation Dk fℓ (0) = Dk f (0) for
4.5. A Generalization to the Case.....
113
every ℓ ≥ 0 and the convergence of ( fℓ ) to f in the space C k (O, Rn ).
As a result and by the continuity of A, the sequence (Mℓ ) and (Nℓ ) (respectively (Deξ Mℓ ) and (Deξ Nℓ )) tend to M and N (respectively Deξ M and
Deξ N) in the space C 0 ([−r, r] × C × △, Rn+1 ) (respectively C 0 ([−r, r] ×
C × △, L (Rn+1 , Rn ))).
Theorem 5.1. Let 0 < γ < 1 be a given constant. After shrinking
δ > 0 if necessary, there exists r > 0 such that the mapping Nℓ (t, e
ζ0 , ·) is
Lipschitz-continuous with constant γ from the ball △ into itself for every
pair (t, e
ζ0 )ǫ[−r, r] × C and every ℓ ≥ 0.
SKETCH OF THE PROOF: Because the result needs to be uniform with 145
respect to the index ℓ, the proof given in Theorem 3.1 cannot be repeated. Showing that ||Deξ Nℓ || is bounded by γ on a neighbourhood of
the point (0, e
ξ0 , e
ξ0 ) in R × C × △ by a continuity argument does not ensure that the neighbourhood in question is independent of ℓ. Instead, we
establish an estimate of ||Deξ Nℓ (t, e
ζ0 , e
ξ)||. In Appendix 2, this estimate is
1
found to be
2||Deξ gℓ (0, e
ξ0 )||
3δ
+
ζ0 , e
ξ)|| ≤
||Deξ Nℓ (t, e
ω(δ)+
(1 − δ)
(1 − δ)3
"

#
 ||A(e
3
ξ0 )|| + ω(δ) 
e
e
e


||gℓ (t, ξ)|| + ||Deξ gℓ (t, ξ) − Deξ gℓ (0, ξ0 )|| ,
+ 2

(1 − δ)
(1 − δ)2
for every (t, e
ζ0 , e
ξ)ǫ[−r, r] × C × △ and every ℓ > 0, where
ω(δ) = sup ||A(e
ζ0 ) − A(e
ξ0 )||.
e
ζ0 ǫC
By the compactness of C and the continuity of A, ω(δ) is well - defined and tends to 0 as δ tends to 0. Using the convergence results mentioned before, it is now possible to parallel the proof of Theorem 3.1 to
get the desired conclusion.
1
Recall that Deξ gℓ (0, e
ξ0 ) = Dq(e
ξ0 ) for every ℓ ≥ 0 but this form is not quite appropriate in the formula.
114
4. An Algorithm for the Computation of the Branches
From Theorem 5.1, the sequence
e
ζℓ+1 = Nℓ (t, e
ζ0 , e
ζℓ ), ℓ ≥ 0,
146
(5.11)
is well-defined in the ball △ but it is less obvious that in §4 than it tends
to a fixed point of the mapping N(t, e
ζ0 , ·) in △.
Note from the uniform convergence of the sequence (Nℓ ) to N and
Theorem 5.1 that the mapping N(t, e
ζ0 , ·) is also Lipschitz-continuous
with constant γ so that it has a unique fixed point e
ξ in △. The problem is
to prove that the sequence (5.11) tends to e
ξ. As is will provide us with
useful information, we now establish this convergence result. Let e
ζ∈△
be any cluster point of the sequence (e
ζℓ ) (whose existence follows from
the compactness of △) and (e
ζℓi ) a subsequence with limit e
ζ. Using the
convergence of Nℓ to N, we shall prove that e
ξ=e
ζ if we show that the
sequence (e
ζℓi ) and e
ζℓi +1 have the same limit. This will follow from the
relation
lim ||e
ζℓ − e
ζℓ+1 || = 0.
(5.12)
ℓ→+∞
Once (5.12) is established, we immediately deduce that the sequence
(e
ζℓ ) has e
ξ as a unique cluster point, whence the convergence of the whole
sequence (e
ζℓ ) to e
ξ follows by a classical result, in topology, for compact
sets.
To prove (5.12), write
e
ζℓ+1 − e
ζℓ = Nℓ (t, e
ζ0 , e
ζℓ ) − Nℓ (t, e
ζ0 , e
ζℓ−1 ) =
= Nℓ (t, e
ζ0 , e
ζℓ ) − Nℓ (t, e
ζ0 , e
ζℓ−1 )+
= +Nℓ (t, e
ζ0 , e
ζℓ−1 ) − Nℓ (t, e
ζ0 , e
ζℓ−1 ).
From Theorem 5.1, we find
147
||e
ζℓ+1 − e
ζℓ || < γ||e
ζℓ − e
ζℓ−1 || + ||Nℓ − Nℓ−1 ||∞,[−r,r]×C×△ .
(5.13)
Due to the convergence of the sequence (Nℓ ) to N, given any ǫ > 0,
one has
||Nℓ − Nℓ−1 ||∞,[−r,r]×C×△ < ǫ,
for ℓ large enough, say ℓ ≥ ℓ0 ≥ 1. From (5.13),
||e
ζℓ+1 − e
ζℓ || ≤ γ||e
ζℓ − e
ζℓ−1 || + ǫ
4.5. A Generalization to the Case.....
115
for ℓ ≥ ℓ0 . Define the sequence (for ℓ ≥ ℓ0 )
aℓ+1 = γaℓ + ǫ
aℓ0 = ||e
ζℓ0 − e
ζℓ0 −1 ||.
By an immediate induction argument, it can be seen that
||e
ζℓ − e
ζℓ−1 || ≤ aℓ
for every ℓ ≥ ℓ0 . But the sequence (aℓ )ℓ≥ℓ0 tends to the real number
a = ǫ/(1 − γ) and, for ℓ ≥ ℓn large enough, say ℓ ≥ ℓ1 , we have
aℓ ≤
Therefore
2ǫ
.
1−γ
2ǫ
||e
ζℓ − e
ζℓ−1 || ≤
,
1−γ
for ℓ ≥ ℓ1 and (5.12) follows, since ǫ > 0 is arbitrarily small.
COMMENT 5.1. Setting
e
xℓ = te
ζℓ ,
(5.14)
||e
xℓ − e
x|| ≤ K|t|(γ′ )ℓ
(5.15)
the sequence (e
xℓ ) tends to e
x = te
ξ, one of the 2ν solutions of the equation 148
f (e
x) = 0 with norm |t|. Here, it is not possible to ascertain that the rate
of convergence of (e
xℓ ) to e
x is geometrical because it depends on the rate
of convergence of the sequence (Nℓ ) to N in the space C 0 ([−r, r] × C ×
△, Rn+1 ), as it appears in (5.13).
However, if this convergence is geometrical too, it is immediate, by
(5.13)-(5.14), that there are constant 0 < γ′ < 1 and K > 0 such that
for every ℓ ≥ 0. The geometrical convergence of the sequence (Nℓ )
to N in the space C 0 ([−r, r] × C × △, Rn+1 ) is ensured if, for instance,
the sequence ( fℓ ) tends to f geometrically in the space C k (O, Rn ). Unfortunately, this assumption is beyond the best result that the natural
hypothesis ‘ f ∈ C k ’ provides in the applications we have in mind, (if
116
4. An Algorithm for the Computation of the Branches
f ∈ C m with m > k, the geometrical convergence in C k (O, Rn ) raises
no problem) namely, the geometrical rate of convergence of ( fℓ ) to f
in the space C k−1 (O, Rn ). If so, it can be shown (cf. Appendix 2) that
the rate of convergence of the sequence (e
xℓ ) to e
x remains geometrical:
More precisely, there are constants 0 < γ′ < 1 and K > 0 such that
||e
xℓ−ex || ≤ K(γ′ )ℓ
(5.16)
for every ℓ ≥ 0 (note the difference with (5.15)).
149
Remark 5.2. Recall that the convergence of the sequence ( fℓ ) to f in
the space C k (O, Rn ) is assumed throughout this chapter. In the result
quoted above, this assumption is not dropped but no special rate of conevergence is required in this space.
As in §4, a scheme for the computation of e
x in terms of the sequence
( fℓ ) can be easily obtained from (5.11) and (5.14): After the choice of
e
x0 = te
ζ0 , the sequence

x̄ℓ



xℓ+1 = |t|
,
 e
|| x̄||ℓ



k!
 e
A(e
ζ0 ) fℓ (e
xℓ ), ℓ ≥ 0.
xℓ = e
xℓ − tk−1
(5.17)
4.6 Application to One-Parameter Problems.
The scheme (5.17) is especially suitable for the computation of the
branches of solution in one parameter problems as defined in general
in Cahpter 1. For the sake of clarity, we shall first describe a more abe be as real Banach space with norm || · ||, B(0, ρ)
stract situation: Let Z
e and Φ(=
the closed ball with radius ρ > 0 centered at the origin in Z
k
e
Φ(e
x,e
z))ǫC (O × B(0, ρ), Z), k ≥ 1 (note that O × B(0, ρ) = O × B(0, ρ)
e = C k (O × B(0, ρ), Z)
e but this not a normed
so that C k (O × B(0, ρ), Z)
e
space when Z is infinite dimensional since B(0, ρ) is not compact) with
Φ(0) = 0,
Dez Φ(0) = 0.
4.6. Application to One-Parameter Problems.
150
117
Given any constant 0 < β < 1 and after shrinking ρ > 0 and the neighbourhood O if necessary, we may assume that
|||Dez Φ(e
x,e
z)||| ≤ β,
for every (e
x,e
z)ǫO × B(0, ρ). Applying the mean value theorem, we see
that the mapping Φ(e
x, ·) is a contraction with constant β from B(0, ρ)
e
into Z. In addition, after possibly shrinking the neighbourhhod O once
xǫO.
again, the mapping Φ(e
x, ·) maps tha ball B(0, ρ) into itself for very e
Indeed, given (e
x,e
z)ǫO × B(0, ρ), we have
|||Φ(e
x,e
z)||| ≤ |||Φ(e
x,e
z) − Φ(e
x, 0)||| + |||Φ(e
x, 0)|||
≤ β|||e
z||| + |||Φ(e
x, 0)|||
≤ βρ + |||Φ(e
x, 0)|||,
and the result follows from the fact that |||Φ(e
x, 0)||| can be made less that
(1 − β)ρ for every e
xǫO by the continuity of Φ at the origin.
Therefore, for every e
xǫO, the sequence



ϕ0 (e
x) = 0
 e
(6.1)


 e
ϕℓ+1 (e
x) = Φ(e
x, ϕℓ (e
x)), ℓ ≥ 0,
e with values
is well defined, each mapping e
ϕℓ being in the space C k (O, Z)
in B(0, ρ) and verifies
e
ϕℓ = 0, ℓ ≥ 0.
The sequence (e
ϕℓ ) also tends (pointwise) to the mapping e
ϕ characterized by
e
ϕ(e
x) = Φ(e
x, e
ϕ(e
x)),
for every e
xǫO. In particular
e
ϕ(0) = 0.
Actually, the mapping e
ϕ is of class C k in O. Indeed, for every
ϕ(e
x) is a solution for the equation
e
xǫO, e
e
z − Φ(e
x,e
z) = 0.
151
4. An Algorithm for the Computation of the Branches
118
As e
ϕ(e
x)ǫB(0, ρ), one has |||Dez Φ(e
x, e
ϕ(e
x))||| ≤ β < 1 and hence
e
I − Deξ Φ(e
x, e
ϕ(e
x))ǫI som(Z).
Thus, the mapping e
ϕ(·) is of class C k around e
x from the Implicit
e
function theroem, which shows that e
ϕǫC k (O, Z).
Naturally, one may expect the convergence of the sequence (e
ϕℓ ) to
be better than pointwise. Indeed, it so happens that the sequence (e
ϕℓ )
e the convergence being geometraical in
tends to e
ϕ in the space C k (O, Z),
e (See Appendix 2).
the space C k−1 (O, Z)
k (O × B(0, ρ), Rn ) be such that
e
Let now F(= F(e
x, Z))ǫC
F(0) = 0.
The mapping
e = F(e
e
xǫO → f ( x)
x, e
ϕ(e
x))ǫRn ,
(6.2)
is of class C k in O and
f (0) = 0.
The same property holds for each term of the sequence
152
e
xǫO → fℓ (e
x) = F(e
x, e
ϕℓ+k (e
x)), ℓ ≥ 0.
(6.3)
D j fℓ (0) = D j f (0), 0 ≤ j ≤ k,
(6.4)
The sequence ( fℓ ) tends to f in the space C k (O, Rn ), the rate of
convergence being geometrical in the space C k−l (O, Rn ). Also
for every ℓ ≥ 0.
Remark 6.1. The definition of the mapping fℓ has been through the term
e
ϕℓ+k instead of the “natural one” e
ϕℓ in order that property (6.4) holds.
More generally, the mapping fℓ can be defined through the term e
ϕℓ+k ,
with k′ ≥ k.
4.6. Application to One-Parameter Problems.
119
By choosing e
x0 = te
ζ0 , the scheme (5.17) becomes

xℓ


,
xℓ+1 = |t| ||x||
 e
ℓ


k!
 xℓ = e
xℓ − tk−1 A(e
ζ0 )F(e
xℓ , e
ϕℓ+k (e
xℓ )).
e
In the above formula, e
ϕℓ+k (e
xℓ ) coincide with the element e
zℓ+k,ℓ ǫ Z
defined for every ℓ ≥ 0 by



z0,ℓ = 0,
 e


 e
z j+1,ℓ = Φ(e
xℓ ,e
z j,ℓ ), 0 ≤ j ≤ ℓ + k − 1.
To sum up, starting from an arbitrary element e
ζ0 ǫS n “close enough”
to e
ξ0 and setting e
x0 = te
ζ0 (|t| > 0 small enough), the sequence (e
xℓ ) is 153
defined by


e
z0,ℓ = 0,







z j+1,ℓ = Φ(e
xℓ ,e
z j,ℓ )0 ≤ j ≤ ℓ + k − 1,
 e
(6.5)


k!
e

x
,e
z
),
x
=
e
x
−

ℓ
ℓ+k,ℓ
ℓ
ℓ
k−1 A(ζ0 ).F(e

t



 e
xℓ+1 = |t| ||xxℓℓ || , ℓ ≥ 0.
The rate of convergence of the sequence (e
xℓ ) is then geometrical.
Remark 6.2. This algorithm is a “two-level” iterative process since
computing the element e
xℓ+1 from the element e
xℓ requires ℓ + k − 1 iterates. Formally, “one-level” algorithms are easy to derive from the
scheme (6.5) but no convergence has been proved yet.
The application to one-parameter problems in Banach spaces is now
an immediate consequence. The notation and hypothesis being the same
as in Chapter 1, it has been shown that finding the local zero set of the
mapping G(= G(e
x)) is equivalent to the system
Q1 G(e
x1 + e
x2 ) = 0,
Q2 G(e
x1 + e
x2 ) = 0.
e2 , Y
e2 ), the system can be rewritten as
Since DexG(0)|Xe2 ǫI som(X
Q1 G(e
x1 + e
x2 ) = 0,
4. An Algorithm for the Computation of the Branches
120
154
e
x2 − (DG(0)|Xe2 )−1 Q2 G(e
x1 + e
x2 ) = e
x2 .
This is nothing but
F(e
x1 , e
x2 ) = 0,
where
Φ(e
x1 , e
x2 ) = e
x2 ,
F(e
x1 , e
x2 ) = Q1G(e
x1 + e
x2 ),
Φ(e
x1 , e
x2 ) = e
x2 − [DG(0)|Xe2 ]−1 Q2G(e
x1 + e
x2 ).
(6.6)
(6.7)
e1 ≃ Rn+1 , Y1 ≃ Rn and setting Z
e = X
e2 , it is immediately
As X
seen that the mappings F and Φ verify the required conditions (and,
of course, the mapping e
ϕ of this section coincides with that of Chapter
1). Dropping the subscript 1 in the variable e
x1 , the scheme (6.5) can be
e1 (on which
written as follows: choosing an inner product in the space X
e
e
depends the unit sphere in X1 ) and setting e
x0 = tζ0 where e
ζ is taken
“close enough” to a given normalised element e
ξ0 at which the mapping
q(e
ξ) = Dk f (0) · (e
ξ)k vanishes ( f being the reduced mapping here), the
sequence (e
xℓ ) is defined by

e2 ,

e
z0,ℓ = 0ǫ X






e2 , 0 ≤ j ≤ ℓ + k − 1,

xℓ + e
z j,ℓ )ǫ X
z j+1,ℓ = e
z j,ℓ − [DG(0)|X2 ]−1 · Q2G(e
 e


k!
e1 ,

ζ0 ) · Q1G(e
xℓ + e
zℓ+k,ℓ )ǫ X
xℓ = e
xℓ − tk−1 A(e





x̃ℓ e
 e
xℓ+1 = |t| ||xℓ || ǫ X1 , ℓ ≥ 0.
(6.8)
155
e1 afRemark 6.3. Since the choice of the inner product in the space X
fects the unit sphere, the mapping A also depends on the particular
choice of the inner product.
COMMENT 6.1. The scheme (6.8) provides an approcimation e
xℓ of the
e
solution e
xǫ X1 to the reduced equation f (e
x) = Q1 G(e
x+e
ϕ(e
x)) = 0 with
norm |t| that we have selected among the 2ν ones. The corresponding
solution to the initial problem (namely e
x +e
ϕ(e
x)) is then approximated by
4.6. Application to One-Parameter Problems.
121
e
xℓ + e
ϕℓ+k (e
xℓ ) which is none other than e
xℓ +e
zℓ+k,ℓ . Besides, the sequence
(e
xℓ + e
zℓ+k,ℓ ) converges to e
x+e
ϕ(e
x) geometrically. Indeed, the sequence
e
(e
xℓ ) tends to e
x geometrically and denoting || · || the norm in the space X
e
(hence in X2 ) one has
|||e
zℓ+k,ℓ − e
ϕ(e
x)||| ≤ |||e
ϕℓ+k (e
xℓ ) − e
ϕ(e
xℓ )||| + |||e
ϕ(e
xℓ − e
ϕ(e
x))|||.
The first term in the right hand side is bounded by the norm |||e
ϕℓ+k −
ϕℓ+k tends e
ϕ
e
ϕ|||∞,O which tends to 0 geometrically, since the sequence e
k−1
0
e2 ) and hence in C (O, X
e2 ). As for
geometrically in the space C (O, X
the second term, it is bounded by |||De
ϕ|||∞,O ||e
xℓ − e
x|| for the mapping e
ϕ
xℓ || = ||e
x||
is at least of class C 1 in O and thus in the ball of radius |t| = ||e
centered at the origin of Rn+1 (more rapidly, one can say that it is not
restrictive to assume that O in convex).
Application to the Problems of Birurcation from the trivial Branch.
When the analysis of Chapter 3, §2 is available, we can apply the
scheme (6.8) to problem of bifurcation from the trivial branch: Find 156
(µ, x)ǫR × X around the origin such that
G(µ, x) = (I − (λ0 + µ)L)x + Γ(µ, x) = 0,
(6.9)
where λ0 is a characteristic value of L with (algebraic and geometric)
multiplicity n > 1. Here, we shall take
X1 = Y1 = Ker(I − λ0 L),
X2 = Y2 = Range(I − λ0 L),
This choice being possible precisely because the algebraic and gee1 = R × X1
ometric multiplicaties of λ0 are the same. It follows that X
e2 = X2 .
while X
Under the assumptions of Chapter 3, §2 and when k = 2, no change
e1 is the pair (µ, x)ǫR×X1
of the parameter µ is required. The variable e
xǫ X
and the mapping q becomes (cf, Chapter 3 §2, relation (2.13))
(µ, x)ǫR × X1 → q(µ, x) = −
2µ
x + Q1 D2x Γ(0) · (x)2 ǫX1 .
λ0
4. An Algorithm for the Computation of the Branches
122
Let
g0 =
zeta
157
(µ0 , x0 )
,
||(µ0 , x0 )||
(6.10)
be “close enough” to some selected element of the zero set of q. Clearly,
we can assume that ||(µ0 , x0 )|| > 0 is as small as desired. Further, by
changing (µ0 , x0 ) into a non-zero collinear element (with small norm)
the definition of e
ζ0 is unchanged, unless the orientation of (µ0 , x0 ) is
reversed and, in such a case, e
ζ0 is transformed into −e
ζ0 . This explains, in
the relation (µ0 , x0 ) = te
ζ0 = ||(µ0 , x0 )||e
ζ0 , that the quantity t is ||(µ0 , x0 )||
and hence positive. In the scheme (6.8) as well as in the general one
(5.17) from which it is derived, it is also possible to limit ourselves to
the case t > 0, provided we change e
ζ0 into −e
ζ0 when t(= ±||e
x0 || in any
case) should change sign. But this did not lead to a simple exposition
at that time, which is the reason why this property was not mentioned
earlier. Thus, starting with the point (µ0 , x0 ) and defining e
ζ0 by (6.10),
the scheme (6.8) will provide the only solution of the reduced equation
of (6.9) which is “close” to (µ0 , x0 ) and has the same norm as (µ0 , x0 ).
The corresponding curve is obtained by moving (µ0 , x0 ) along the line it
generates in R × X1 . Denoting by z (instead of e
z) the generic variable of
e2 ), the explicit form of the scheme (6.8) is
the space X2 (= X



















z0,ℓ = 0ǫX2 ,
z j+1,ℓ = [(I − λ0 L)|X2 ]−1 · [−µQ2 Lz j,ℓ + Q2 Γ(µℓ , xℓ + z j,ℓ )]ǫX2 , 0 ≤ j ≤ ℓ + 1,
h
i
ζ0 ) · − λµ0ℓ xℓ + Q1 Γ(µℓ , xℓ + zℓ+2,ℓ ) ǫR × X1 ,
(µℓ , xℓ ) = (µℓ , xℓ ) − ||(µ02,x0 )|| A(e
(µℓ ,xℓ )
ǫR × X1 , ℓ ≥ 0.
(µℓ+1 , xℓ+1 ) = ||(µ0 , x0 )|| ||(µ
ℓ ,xℓ )||
(6.11)
According to Comment 6.1, the approximation of the corresponding
solution of the equation (6.9) associated with the iterate (µℓ , xℓ ) is
(µℓ , xℓ + zℓ+2,ℓ ).
158
Under the assumption of Chapter 3, § 2 and when k ≥ 3, a preliminary change of parameter µ into ηk−1 when k is even, into σηk−1 ,
σ = ±1 when k is odd, is necessary. We shall consider the case k even
only, leaving it to the reader to make the obvious modification when k
e1 is the pait (η, x)ǫR × X1 and the mapping q
is odd. The variable e
xǫ X
4.6. Application to One-Parameter Problems.
123
becomes (cf, Chapter 3, §2)
(η, x)ǫR × X1 → q(η, x) = −
Let
k!ηk−1
x + Q1 Dkx Γ(0) · (x)k ǫX1
λ0
(η0 , x0 )
e
,
ζ0 =
||(η0 , x0 )||
be close enough to some selected element of the zero set of q. Taking
||(η0 , x0 )|| sufficiently small and for the same reasons as when k = 2, the
explicit form of the scheme (6.8) we obtain is (denoting by z instead of
e2
e
z) the variables of the space X2 = X

































z0,ℓ = 0ǫX2 ,
z j+1,ℓ = [(I − λ0 L)|X2 ]−1 · [−ηℓk−1 Q2 Lz j,ℓ + Q2 Γ(ηℓk−1 , xℓ + z j,ℓ )]ǫX2 ,
0 ≤ j ≤ ℓ + k − 1,
ηk−1
k!
ℓ
e
xℓ + Q1 Γ(ηk−1 , xℓ + zℓ+k,ℓ )]
(ηℓ , xℓ ) = (ηℓ , xℓ ) −
k−1 A(ζ0 ) · [−
||(ηℓ ,xℓ )||
(ηℓ+1 , xℓ+1 ) =
(ηℓ ,xℓ )
||(η0 , x0 )|| ||(η
ǫR
ℓ ,xℓ )||
λ0
× X1 , ℓ ≥ 0.
ℓ
ǫR × X1 ,
(6.12)
According to Comment 6.1, the approximation of the corresponding
solution of the equation (6.9) in which µ is replaced by ηk−1 is
159
(ηℓ , xℓ + zℓ+k,ℓ ).
(6.13)
Thus, the approximation of the corresponding solution of equation
(6.9) itself is
(ηk−1
ℓ , xℓ + zℓ+k,ℓ ),
whose convergence is geometrical (since that of the sequence (6.13)) is
geometrical from Comment (6.1).
Remark 6.4. On comparison with the classical Lyapunov-Schmidt
Method, the above algorithm has the disadvantage of being a “two level” iterative process (cf. Remark 6.2). However, it has two basic
advantages: The first one is that it does not require the zero set of the
124
160
4. An Algorithm for the Computation of the Branches
mapping q to be known exactly, an assumption unrealistic in the applications. The Lyapunov-Schmidt method, which consists in finding the
curves of solutions tangent to any given line in the zero set of q by an
iterative process uses this particular line explicitly (see e.g. [2]). Of
course, one can expect that the Lyapunov-Schmidt method will neverthless be satisfactory if this line is replaced by a good approximation to it
(though we are not aware of a proof of this). In the process described
above, a rough estimate (represented by e
ζ0 ) is sufficient because the algorithm makes the corrections itself. It then appears more “stable” than
the Lyapunov-Schmidt method.
The second advantage is obvious in so far as the schemes (6.11)(6.12) are particular cases of the general algorithm (6.8) that can be used
when no branch of solutions is known a priori, while the LyapunovSchmidt method is limited to problems of bifurcation from the trivial
branch or, at the best, to problems in which on of the branches is known
explicitly. We leave it to the reader to derive from (6.8) an iterative
scheme associated with the problem of Chapter 3 § 3 (in which no
change of parameter is required in practice).
Chapter 5
Introduction to a
Desingularization Process in
the Study of Degenerate
Cases
LET f : Rn+1 → Rn BE as in Chapter 2, namely such that the first k − 1 161
derivatives of f vanish at the origin and the kth does not (k > 1). In
chapter 2, we have solved the problem of finding the local zero set of f
when the polynomial mapping
e
ξǫRn+1 → q(e
ξ) = Dk f (0) · (e
ξ)k ǫRn ,
verifies the condition (R − N.D.). The next step consists in examining
what happens when the mapping q does not verify the nondegeneracy
condition in question and the aim is to find hypotheses as general as
possible ensuring that the local zero set of f is still a finite union of
curves through the origin. Problems in which the associated mapping q
does not verify the condition (R−N.D.) will be referred to as degenerate.
In the examples studied in Chapter 3, we have already encountered
degenerate problems but, in the particular context of problems of bifucation from the trivial branch, we were able to overcome the difficulty
125
126
5. Introduction to a Desingularization Process....
by performing a suitable change of the parameter. However, with or
without change of the parameter, the condition
X = Ker(I − λ0 L) ⊕ Range(I − λ0 L),
162
163
namely, the equality of the algebraic and geometric multiplicities of the
characteristic value λ0 , was always seen to be necessary for the condition (R − N.D.) to hold. The failure of the above condition is then
sufficient for the problem to be degenerate and no solution to it has been
found by changing the parameter.
Before any attempt to develop a general theory, it is wise to examine
a few particular cases. The most elementary example is when k = 1:
The mapping q is the linear mapping D f (0) and it does not verify the
condition (R − N.D.) if and only if D f (0)ǫL (Rn+1 , Rn ) is not onto. This
implies that the zero set of q, here Ker D f (0), is a subspace of Rn+1 with
dimension at least two. Hence, it does not consist of a finite number of
lines, the very argument that allowed us to characterize the local zero
set of f as a finite union of curve in Chapter 2. Note however that the
somewhat disappointing situation we have just described cannot occur
if n = 1, for, otherwise, the derivative D f (0) would vanish identically
and k could not be 1. More generally and regardless of the (finite) value
of k, it has been shown when n = 1 that the zero set of q is always
the union of at most k lines through the origin, though the condition
(R − N.D.) holds if and only if the real roots of a certain polynomial in
one variable with real coefficients are simple. Acutally, the examination
of the general case when n is arbitrary but k ≥ 2 shows that the failure
of the condition (R − N.D.) does not necessarily imply that the zero set
of q is made of infinitely many lines. In other words, it is reasonable to
expect the local zero set of f to be made of a finite number of curves
under hypotheses more general than those of Chapter 2.
Assuming then that the zero set of q is made of a finite number of
lines and selecting an element e
ξǫRn+1 − {0} such that q(e
ξ) = 0 and Dq(e
ξ)
is not onto, the analysis of the structure of the local zero set of f “along”
the line Re
ξ is often referred to as the “analysis of bifurcation near the
degenerate eigenray Re
ξ”. The almost inevitable tool here seems to be
the topological degree theory (see e.g. Shaerer [39]) but the information
5.1. Formulation of the Problem and Preliminaries.
127
obtained is necessarily vague and of an essentially theoretical character.
In non-linear eignevalue problems, another approach, more technical,
has been developed by Dancer [8]. Again, the results lose most of their
accuracy (on comparison with the nondegenerate case) and the method
does not seem to have been extended to other problems.
In this chapter, we present an analytic approach which provides results as precise as in the nondegenerate case. The study is limited to
the - apparently - simple situation when n = 1 and k = 2. The general
theory is still imcomplete but the same method can almost readily be
employed and will provide quite precise informations in each particular
problem. The open questions that remain in the general case are internal
to the method and to its links to singularity theory1 (when n = 1 and
k = 2, see Rabier [32]). Also, only a few of the theoretical results that 164
could be derived from it have been investigated so far. As an example of
such a result, we give a statement that complements Krasnoselskii’s theorem (Theorem 1.2 of Chapter 1) in a particular case as well as Crandall
and Rabinowitz’s study of bifurcation from the trivial branch at a simple
characteristic value.
In contrast to Chapter 2, we shall not here bother about formulating the weakest possible regularity assumptions: The mapping f under
study will be of class C ∞ although this hypothesis is clearly unnecessary
in most of the chapter. The exposition follows Rabier [32].
5.1 Formulation of the Problem and Preliminaries.
Let f be a C ∞ real-valued mapping of two variables defined on a neighbourhood of the origin such that f (0) = 0, D f (0) = 0 and det D2 f (0) ,
0. Then, the Morse lemma (Theorem 3.1, 1 and 2 of Chapter 1) states
that the local zero set of f reduces to the origin if det D2 f (0) > 0 and is
made of exactly two curves of class C ∞ intersecting transversally at the
origin if det D2 f (0) < 0. But it does not say anything if det D2 f (0) = 0.
If this determinant vanishes because D2 f (0) = 0, we can examine higher
order derivatives of f at the origin and the structure of the local zero set
1
However, signficant progress has since been made.
128
165
5. Introduction to a Desingularization Process....
of f may follow from an application of Theorem 3.1 of Chapter 2 with
n = 1. If det D2 f (0) = 0 but D2 f (0) , 0, no information is available
and, indeed, elementary examples show that the situation may differ
considerably: Denoting by (u, v) the variable in the plane R2 , the mappings
f (u, v) = u2 + v4 ,
f (u, v) = u2 − v4
f (u, v) = (u − v)2 ,
all verify the condition f (0) = 0, D f (0) = 0, det D2 f (0) = 0 and
D2 f (0) , 0. Their local zero sets are, respectively: the origin, the union
of the two tangent parabolas u = v2 and u = −v2 , and the line u = v.
There three cases are by no means exhaustive, as it is seen with the
mapping
!
1 −1/u2
2
f (u, v) = v − sin
e
.
u
whose local zero set is picture in Figure 1.1 below.
Figure 1.1:
166
This explains why it is not possible to find an exhaustive classification of the local zero sets of such mappings in geometrical terms.
Nevertheless, a partial classification if possible. It will be obtained by
the following. Given a mapping f verifying f (0) = 0, D f (0) = 0, det
D2 f (0) = 0 and D2 f (0) , 0, we shall construct a new mapping f (1)
5.1. Formulation of the Problem and Preliminaries.
129
whose local zero set provides that of f through an elementary transfomation. About the mapping f (1) , the following three possiblities are
exhaustive:
1) D f (1) (0) , 0 so that the local zero set of f (1) can be found through
the Implicit function theorem.
2) D f (1) (0) = 0, det D2 f (1) (0) , 0 so that the local zero set of f (1)
can be found through the Morse lemma.
3) D f (1) (0) = 0, det D2 f (1) (0) = 0 and D2 f (1) (0) , 0 so that finding
the local zero set of f (1) reduces to finding the local zero set of a
new iterate f (2) .
In so doing, we define a sequence of iterates f (1) , · · · , f (m) : The
iterate f (m) exists under the necessary and sufficient condition that the
local zero set of f (m−1) cannot be found through the Implicit function
theorem or the Morse lemma. If the process ends after a finite number
of steps, it corresponds to a desingularization of the initial mapping f .
If it is endless, there are two possibilities, among which one of them
bears a geometric characterization (see § 8).
We shall complete this section by giving a definition and establishing a preliminary property. Of course, the mapping f will henceforth 167
verify the condition f (0) = 0 and
D f (0) = 0,
(1.1)
detD2 f (0) = 0,
(1.2)
2
(1.3)
D f (0) , 0.
In particular, (1.2) means that the null-space of the mapping D2 f (0)
ǫL (R2 , L (R2 , R)) does not reduce to {0}. Because of (1.3), this nullspace is not the whole space R2 and it is then one-dimensional.
Definition 1.1. The one-dimensional space
Ξ = Ker D2 f (0),
will be called the characteristic of f .
5. Introduction to a Desingularization Process....
130
The following elementary result will be used several times.
Lemma 1.1. Under our assumptions, D2 f (0) · (e
ξ)2 = 0 for some e
ξǫR2
if and only if e
ξǫΞ.
Proof. It is obvious that D2 f (0) · (e
ξ)2 = 0, if e
ξǫΞ. Conversely, let e
ξǫR2
2
2
e
be such that D f (0) · (ξ) = 0 and assume by contradiction that there is
e
τǫR2 with D2 f (0) · (e
ξ,e
τ) , 0. Necessarily, e
ξ and e
τ are not collinear, so
2
e
that {ξ,e
τ} is a basis of R . With respect to this basis, D2 f (0) becomes
"
168
#
0
D2 f (0) · (e
ξ,e
τ)
D2 f (0) · (e
ξ,e
τ) D2 f (0) · (e
τ)2
and its determinant is −(D2 f (0) · (e
ξ,e
τ))2 < 0, which contradicts the
2
hypothesis that it vanishes .
5.2 Desingularization by Blowing-up Procedure.
We shall begin with an approach we have already used in Chapter 2.
Extending (in theory) the mapping f to the whole space R2 as a C ∞
mapping, we first reduce the problem of finding the non-zero solutions
of the equation f (e
x) = 0 with prescribed euclidian norm ||e
x|| to the problem



x)ǫ(R − {0}) × S 1 ,
 (t, e
(2.1)


 g(t, e
ξ) = 0,
where the mapping g defined on R × R2 with values in R2 is given by

1


ξ) if t , 0,
 t2 f (te
g(t, e
ξ) = 

 1 D2 f (0) · (e
ξ)2 if t = 0.
(2.2)
2
The relationship of the nonzero solutions e
x of the equation f (x) = 0
to the solutions (t, e
ξ) of the equation g(t, e
ξ) = 0 is of course that e
x can
always be written as e
x = te
ξ (in two different ways).
5.2. Desingularization by Blowing-up Procedure.
169
131
Remark 2.1. The mapping g in (2.2) differs from that of Chapter 2 by
the multiplicative factor 1/2 only. This modification will bring some
simplifications in the formulas later.
Writing the Taylor expansion of f about the origin, we find that
g(t, e
ξ) =
Z1
0
(1 − s)D2 f (ste
ξ) · (e
ξ)2 ds,
(2.3)
for every (t, e
ξ)ǫR × R2 , which shows that the mapping g is of class C ∞ .
The next result, very similar to Lemma 3.2 of Chapter 2, is based on
the observation that the intersection S 1 ∩ Ξ reduces to two points. For
the sake of convenience, we shall say that two problems are equivalent
if all the solutions of one of them provide all the solutions of the other.
We leave it to the reader to keep track of the (obvious) correspondence
between the solutions of the problems under consideration.
Lemma 2.1. Let e
ξ0 be either point of the intersection Ξ ∩ S 1 and C
a given neighbourhood of e
ξ0 in S 1 . Then, for r > 0 small enough the
problems



ξǫS 1 ,
 0 < |t| < r, e
(2.4)


 g(t, e
ξ) = 0,
and
are equivalent.



ξǫC,
 0 < |t| < r, e


 g(t, e
ξ) = 0,
(2.5)
Proof. First, note that the problem (2.5) is equivalent to



ξǫC ∪ (−C)
 0 < |t| < r, e


 g(t, e
ξ) = 0,
170
(2.6)
2
Recall that the condition det D2 f (0) = 0 is independent of the choice of any basis
in R2 .
132
5. Introduction to a Desingularization Process....
since each solution of (2.5) is obviously a solution of (2.6) and conversely, given any solution (t, e
ξ) of (2.6), either (t, e
ξ) or (−t, −e
ξ) is a
solution of (2.5) from the relation g(t, e
ξ) = g(−t, −e
ξ). We shall prove the
assertion by showing that the problems (2.4) and (2.6) have the same
solutions for r > 0 small enough. Each solution of (2.6) is clearly a
solution of (2.4). Conversely, assume, by contradiction, that there is no
r > 0 such that each solution of (2.4) is a solution of (2.6). This means
that there is a sequence (tℓ , e
ξℓ )ℓ≥1 with lim tℓ = 0, e
ξℓ ǫS 1 C ∪ (−C) and
ℓ→+∞
g(tℓ , e
ξℓ ) = 0. From the compactness of S 1 , we may assume that there
e
is ξǫS 1 such that e
ξℓ converges to e
ξ. By the coninuity of g, we deduce
e
g(0, ξ) = 0. But, by definition of g(0, ·) and Lemma 1.1, e
ξ must be one
of the elements e
ξ0 or −e
ξ0 , which is impossible, since e
ξℓ < C ∪ (−C) for
every ℓ ≥ 1, by hypothesis.
Let us now consider a subspace T of R2 such that
R2 = −Ξ ⊕ T.
(2.7)
Given a non-zero element e
τ0 ǫT , we may write T = Re
τ0 . Let Φ :
R2 → R × S 1 be the C ∞ mapping defined by
171
Φ(u1 , v1 ) = (u1 ||e
ξ0 + v1e
τ0 ||, (e
ξ0 + v1e
τ0 )/||e
ξ0 + v1e
τ0 ||),
(2.8)
(where | · | still denotes the euclidean norm). Clearly, Φ(0, 0) = (0, e
ξ0 )
and an elementary calculation yields
DΦ(0, 0) · (u1 , v1 ) = (u1 , v1e
τ0 − v1 (e
ξ0 |e
τ0 )e
ξ0 ),
for (u1 , v1 )ǫR2 , where (·|·) denotes the inner product of R2 . The above
relation shows that the mapping DΦ(0, 0) is one-to-one (as an element
of L (R2 , R3 )) and hence is an isomorphism of R2 to R × Teξ0 S 1 (= R ×
{e
ξ0 }⊥ ). Thus, Φ is a C ∞ diffeomorphism between a neighbourhood of
the origin in R2 and a neighbourhood of the point (0, e
ξ0 ) in R × S 1 . We
can suppose that the latter contains the product (−r, r) × C of Lemma
2.1: it suffices to choose C small enough and shrink r > 0 accordingly.
5.2. Desingularization by Blowing-up Procedure.
133
Then, Φ−1 ((−r, r) × C) is a neighbourhood W1 of the origin in R2 and
each pair (t, e
ξ)ǫ(−r, r)× has the form



ξ0 + v1e
τ0 ||,
 t = u1 ||e


 e
ξ = (e
ξ0 + v1e
τ0 )/||e
ξ0 + v1e
τ0 ||,
with (u1 , v1 )ǫW1 . As t = 0 if and only if u1 = 0 and from lemma 2.1.
our initial problem of finding the nonzero solutions e
x in the local zero
set of f is equivalent to



 (u1 , v1 )ǫW1 , u1 , 0,


 g(u1 ||e
ξ0 + v1e
τ0 ||, (e
ξ0 + v1e
τ0 )/||e
ξ0 + v1e
τ0 ||) = 0
Coming back to the definition of g ((2.2)), this also reads



(u1 , v1 )ǫW1 , u1 , 0,



 f (u1 (e
ξ0 + v1e
τ0 )) = 0.
Finally, multipying by 1/u21 , the problem becomes



 (u1 , v1 )ǫW1 , u1 , 0,


 g(u1 , e
ξ0 + v1e
τ0 ) = 0.
We shall denote by f (1) the C ∞ real-valued function of two variables
f (1) : (u1 , v1 )ǫR2 → f (1) (u1 , v1 ) = g(u1 , e
ξ0 + v1e
τ0 )


1

ξ0 + u1 v1e
τ0 ) if u1 , 0,

 u21 f (u1e
=

 1 v2 D2 f (0) · (e

ξ0 )2 if u1 = 0.
2 1
(2.9)
From the above comments, the nonzero solutions e
x of the equation
f (e
x) with ||x|| < r are of the form
e
x = u1e
ξ0 + u1 v1e
τ0
(2.10)
with (u1 , v1 )ǫW1 , u1 , 0 and f (1) (u1 , v1 ) = 0. Obviously, the characterization (2.10) remains true with the solution e
x = 0 since f (1) (0, 0) = 0.
172
5. Introduction to a Desingularization Process....
134
In other words, since the neighbourhood W1 = Φ−1 ((−r, r) × C) can be
taken arbitrarily small (by shrinking C and r > 0), the problem reduces
to finding the local zero set of f (1) , from which the local zero set of is
deduced through the transformation (2.10).
173
Remark 2.2. The reader may wonder why we have not defined the mapping f (1) by formula (2.9) directly. Indeed, it is clear that every element
of the form (2.10) with (u1 , v1 ) in the local zero set of f (1) is in the local
zero set of f . But the problem is to show that every element in the local
zero set of f is of the form (2.10) with (u1 , v1 ) in the local zero set of
f (1) , a fact that uses the compactness of the unit circle S 1 , essential in
Lemma 2.1.
COMMENT 2.1. The vector e
ξ0 ǫΞ has been chosen so that e
ξ0 ǫS 1 i.e.
e
e
||ξ0 || = 1. This is without importance and ξ0 can be taken as an arbitrary
non-zero element of the characteristic Ξ (bu changing the inner product
of R2 for instance). From now on, we shall allow this more general
choice in the definition of the mapping f (1) in (2.9).
All the derivatives of the mapping f (1) can be explicitly computed
in terms of the derivatives of f at the origin. First from (2.9), for every
pair ( j, ℓ)ǫN × N
j
Now
ξ ℓ )(0, e
ξ0 ) · (e
τ0 )ℓ .
(∂ j+ℓ f (1) /∂u1 ∂vℓ1 )(0) = (∂ j+ℓ g/∂t j ∂e
Lemma 2.2. For every pair ( j, ℓ)ǫN × N, one has







j+ℓ
j eℓ
e
(∂ f g/∂t ∂ξ )(0, ξ) = 





174
for every e
ξǫR2 .
0 if ℓ > j + 2
[ j!/( j + 2 − ℓ)!]D j+2 f (0) · (e
ξ) j+2−ℓ (2.11)
if 0 ≤ ℓ ≤ j + 2,
Proof. Using the expression (2.3) of g and for every pair (t, e
ξ)ǫR × R2 ,
5.3. Solution Through the Implict Function....
135
we find
(∂ g/∂t )(t, e
ξ) =
j
j
Z1
0
(1 − s)s j D j+2 f (ste
ξ) · (e
ξ) j+2 ds.
In particular, for t = 0,
(∂ j g/∂t j )(0, e
ξ) =
and (2.11) follows.
−1
D j+2 f (0) · (e
ξ) j+2 ,
( j + 1)( j + 2)
With Lemma 2.2, we obtain for every pair ( j, ℓ)ǫN × N,



0 if ℓ > j + 2,




j ℓ
j+ℓ (1)
(∂ f /∂u1 ∂v1 )(0) = 
[ j!/( j + 2 − ℓ)!]D j+2 f (0) · ((e
ξ0 ) j+2−ℓ ,





(e
τ0 )ℓ ) if 0 ≤ ℓ < j + 2.
(2.12)
In particular, taking j = 0 and ℓ = 2 in formula (2.12),
(∂2 f (1) /∂v21 )(0) = D2 f (0) · (e
τ0 )2 , 0,
(2.13)
since e
τ0 < Ξ (cf. Lemma 1.1).
5.3 Solution Through the Implict Function Theorem : CUSP Bifurcation
Applying fromula (2.12) with j = 1, ℓ = 0 and j = 0, ℓ = 1 successively,
we get
1
ξ0 )3
(3.1)
(∂ f (1) /∂u1 )(0) = D3 f (0) · (e
6
and
1
ξ0 ,e
τ0 ).
(∂ f (1) /∂v1 )(0) = D2 f (0) · (e
2
But, as e
ξ0 ǫΞ, D2 f (0) • e
ξ0 = 0. Hence
175
(∂ f (1) /∂v1 )(0) = 0.
(3.2)
5. Introduction to a Desingularization Process....
136
From (3.1)-(3.2), the Implict function theorem can be used for finding the local zero set of f (1) if and only if
D3 f (0) · (e
ξ0 )3 , 0,
(3.3)
a condition which is independent of the nonzero element e
ξ0 ǫΞ and of the
(1)
space T chosen for defining the mapping f . Assume then that (3.3)
holds. The Implict function theorem states that the local zero set of f (1)
is made of exactly one curve of class C ∞ . Due to (3.2), this curve is
tangent to the v1 -axis at the origin. In other words, around the origin
f (1) (u1 , v1 ) = 0 ⇔ u1 = u1 (v1 ), u1 (0) = (du1 /dv1 )(0) = 0.
This result can be made more precise by showing that (d2 u1 /dv21 )
(0) , 0: Differentiating two times the identity f (1) (u1 (v1 ), v1 ) = 0 and
since (du1 /dv1 )(0) = (∂ f (1) /∂v1 )(0) = 0, we obtain
(d2 u1 /dv21 )(0) = −
(∂2 f (1) /∂v21 )(0)
(∂ f (1) /∂u1 )(0)
= −6
D2 f (0) · (e
τ0 )2
, 0,
D3 f (0) · (e
ξ0 )3
(3.4)
as it follows from (2.13). In the (u1 , v1 )-plane, the graph of the function
u1 (·) is then as in Figure 3.1
Figure 3.1: Local zero set of f (1)
176
After changing the vector e
ξ0 into −e
ξ0 if necessary, we may assume
(cf. (3.4))
(d2 u1 /dv21 )(0) > 0.
(3.5)
5.3. Solution Through the Implict Function....
137
Theorem 3.1. The vector e
ξ0 being chosen so that (3.5) holds, let us set
Ξ+ = R+e
ξ0 . Then, the local zero set of f is made up of two continuous
half-branches emerging from the origin in the half space Ξ+ ⊕ T . These
half-branches are of class C ∞ away from the origin and tangent to the
characteristic Ξ = Re
ξ0 at the origin. More precisely, there exists an
∞
origin-preserving C diffeomorphism ψ in R such that the local zero set
of f is made up of the two half-branches (ρ > 0 small enough)
177
Proof. After changing the vector e
τ0 into a scalar multiple, we may assume that (d2 u1 /dv21 )(0) = 2 (cf. (3.4)). Not changing e
τ0 introduces
positive multiplicative constants in the expressions below without (of
course) affecting the final result. Thus
u1 (v1 ) = v21 (1 + R(v1 )),
(3.7)
where R is a C ∞ function such that R(0) = 0. The mapping
ϕ(v1 ) = v1 (1 + R(v1 ))1/2 ,
(3.8)
is well defined and of class C ∞ around the origin. In addition
ϕ(0) = 0,
dϕ
(0) = 1.
dv1
It follows that ϕ is an origin-preserving C ∞ local diffeomorphism of
R and the relation (3.7) is simply
u1 (v1 ) = (ϕ(v1 ))2 .
From our choice of ϕ(3.8), we see that
p
ϕ(v1 ) = u1 (v1 ) if v1 > 0,
p
ϕ(v1 ) = − u1 (v1 ) if v1 < 0.
Setting ψ = ϕ−1 , we find
p
v1 = ψ( u1 (v1 )) if v1 ≥ 0.
p
v1 = ψ(− u1 (v1 )) if v1 ≤ 0.
178
5. Introduction to a Desingularization Process....
138
As u1 (v1 ) runs over some interval [0, ρ) for v1 around the origin. It
is equivalent to saying, for every u1 ǫ[0, ρ[, that the two values v1 with
√
u1 = u1 (v1 ) are given by v1 = ±ψ( u1 ).
According toi the general discussion before, the local zero set of f
is made up of elements of the form
e
x = u1e
ξ0 + u1 v1e
τ0 ,
with (u1 , v1 ) in the local zero set of f (1) . Here, we have then
√
τ0
e
x=e
x(α) (u1 ) = u1 e
x0 + u1 ψ((−1)α u1 )e
for u1 ǫ[0, r). Dividing by u1 , 0, we get
√
e
x(α) (u1 ) e
τ0 = e
ξ0
= ξ0 + lim ψ((−1)α u1 )e
u1 →0
u1 →0
u1
lim
This relation expresses that the two half-branches are tangent to
√
the characteristic Ξ at the origin. They are distinct since ψ( u1 ) and
√
ψ(− u1 ) have opposite signs.
From Theorem 3.1, the local zero set of f is then given by Figure
3.2 below.
Figure 3.2: Local zero set of f .
179
Setting e
x = (u, v), the simplest example of mapping f for which the
5.4. Solution Through the Morse Lemma....
139
hypothesis of this section are fulfilled is f (u, v) = u2 ± v3 (or v2 ± u2 , or,
with θ any fixed real number, (v cos θ − u sin θ)2 ± (u cos θ + v sin θ)3 ).
This kind of bifurcation is observed in chemical reaction problems (cf.
Golubitsky and Keyfitz [13]).
5.4 Solution Through the Morse Lemma : Isolated
Solution and Double Limit Bifurcation.
As we saw in the previous section, a necessary and sufficient condition
for the local zero set of the mapping f (1) (2.9) to be found through the
Implicit function theorem is D3 f (0) · (e
ξ0 )3 , 0. Here, we shall then 180
assume
D3 f (0) · (e
ξ0 )3 = 0.
(4.1)
As a result, from (3.1) and (3.2)
(∂ f (1) /∂u1 )(0) = (∂ f (1) /∂v1 )(0) = 0.
(4.2)
Let us now examine the second derivative.Taking j = 2ℓ = 0 and
j = 1, ℓ = 1 successively in (2.12), we find
1 4
D f (0) · (e
ξ0 )4 ,
12
1
ξ0 )3 ,e
τ0 )
(∂2 f (1) /∂u1 ∂v1 )(0) = D3 f (0) · ((e
2
(∂2 f (1) /∂u21 )(0) =
(4.3)
(4.4)
and we already know that (cf. (2.13))
(∂2 f (1) /∂v21 )(0) = D2 f (0) · (e
τ0 )2 , 0.
(4.5)
Therefore, the structure of the local zero set of the mapping f (1) can
be found through the Morse lemma if and only if the determinant det
D2 f (1) (0) is , 0. From the above, this becomes
detD2 f (1) (0) =
1 4
1
(D f (0) · (e
ξ0 )4 )(D2 f (0) · (e
τ0 )2 ) − (D3 f (0) · (e
ξ0 )2 ,e
τ0 )2 , 0.
12
4
(4.6)
5. Introduction to a Desingularization Process....
140
Clearly, det D2 f (1) (0) being non-zero or not is independent of the
choice of the non-zero elements e
ξ0 ǫΞ and e
τ0 ǫT . It is true, but not as
obvious, that our assumptions are independent of the complement T of
the characteristic Ξ chosen for defining the mapping f (1) . We prove this
in
Proposition 4.1. The condition: D f (1) (0) = 0 and det D2 f (1) (0) , 0 is
independent of the choice of the complement T of the characteristic Ξ.
181
Proof. We already know that the condition D f (1) (0) = 0 (i.e. (4.1)) is
independent of T . Assuming D f (1) (0) = 0, we shall see that the condition det D2 f (1) (0) , 0 is independent of T too (see Remark 4.1 below).
This can be checked directly on the formula (4.6) but we prefer here to
give the reason why this independence is true. The second derivative
D2 f (0, e
ξ) of the mapping g ((2.2)) is an element of L (R × R2 , L (R ×
2
R , R)) and our assertion will follow from the equivalence
detD2 f (1) (0) , 0 ⇔ Ker D2 g(0, e
ξ0 ) = {0} × Ξ,
since Ker D2 g(0, e
ξ0 ) is of course independent of any choice of T .
Let then (t, e
ξ)ǫR×R2 be in the null-space Ker D2 g(0, e
ξ0 ). This means



ξ0 ) + (∂2 g/∂t∂e
ξ)(0, e
ξ) · e
ξ = 0ǫR,
 t(∂2 g/∂t2 )(0, e
(4.7)


2
2
 t(∂2 g/∂t∂e
e
e
e
ξ)(0, ξ0 ) + (∂ g/∂ξ )(0, ξ0 ) · e
ξ = 0ǫL (R2 , R)
From Lemma 2.2,
1 4
(∂2 g/∂t2 )(0, e
ξ0 ) =
D f (0) · (e
ξ0 )4 ǫR,
12
1
ξ0 )2 ǫL (R2 , R),
(∂2 g/∂t∂e
ξ)(0, e
ξ0 ) = D3 f (0) · (e
2
(∂2 g/∂e
ξ 2 )(0, e
ξ0 ) = D2 f (0)ǫL (R2 , L (R2 , R)).
182
(4.8)
(4.9)
(4.10)
Since e
ξ0 ǫΞ, one has D2 f (0) · e
ξ0 = 0ǫL (R2 , R). From (4.1) and the
above relations, the equations (4.7) are satisfied with t = 0 and e
ξ =e
ξ0 ,
2
which shows that the line {0} × Ξ is in the null-space Ker D g(0, e
ξ0 ).
To prove that thie line is the whole null-space and because the system
5.4. Solution Through the Morse Lemma....
141
{e
ξ0 ,e
τ0 } is a basis of R2 , it suffices to show with e
ξ = λe
τ0 , λǫR, that the
equations (4.7) have no solution other than t = λ = 0. With (4.8)-(4.10),
they become

t


ξ0 )4 + λ2 D3 f (0) · ((e
ξ0 )2 ,e
τ0 ) = 0ǫR,
 12 D4 f (0) · (e
(4.11)


2
2
 t D3 f (0) · (e
ξ0 ) + λD f (0) · e
τ0 = 0ǫL (R2 , R).
2
The second equation (4.11) is equivalent to two scalar equations,
obtained, for instance, by expressing that the value of the left hand side
vanishes at each of the two noncollinear vectors e
ξ0 and e
τ0 . By the same
arguments as above (i.e. (4.1) and e
ξ0 ǫΞ)the first scalar equation is the
trivial one 0 = 0. The second equation is
t 3
D f (0) · ((e
ξ0 )2 ,e
τ0 ) + λD2 f (0) · (e
τ0 )2 = 0.
(4.12)
2
But the system made of (4.12) and the first equation (4.11) has the
unique solution t = λ = 0 if and only if det D2 f (1) (0) , 0 (cf. (4.6)),
which completes the proof.
ξ0 )3 , 0), it is not true that
Remark 4.1. If D f (1) (0) , 0 (i.e. D3 f (0) · (e
the condition det D2 f (1) (0) , 0 is independent of the choice of T , as the
reader can easily check on formula (4.6).
Remark 4.2. We could already make Proposition 4.1 more precise by
showing that the sign of det D2 f (1) (0) is independent of the choice of T 183
(when D f (1) (0) = 0) but this will be an immediate consequence of our
further analysis.
Theorem 4.1. Assume that D f (1) (0) = 0.
(i) If det D2 f (1) (0) > 0, i.e.
1 4
(D f (0) · (e
ξ0 )4 )(D2 f (0) · (e
τ0 )2 ) − (D3 f (0) · ((e
ξ0 )2 ,e
τ0 ))2 > 0,
3
the local zero set of f reduces to the origin.
(ii) If det D2 f (1) (0) < 0, i.e.
1 4
(D f (0) · (e
ξ0 )4 )(D2 f (0) · (e
τ0 )2 ) − (D3 f (0) · ((e
ξ0 )2 ,e
τ0 ))2 < 0,
3
142
5. Introduction to a Desingularization Process....
the local zero set of f consists of exactly two distinct curves of
class C ∞ , tangent to the characteristic Ξ at the origin. More precisely, there exist two real-valued functions v(α)
1 (u1 ), α = 1, 2, de∞
fined and of class C around the origin, verifying
dv(1)
1
du1
(0) ,
dv(2)
1
du1
(0),
(4.13)
such that the local zero set of f is made up of the two curves (ρ > 0
small enough)
τ0 , α = 1, 2.
u1 ǫ(−ρ, ρ) → e
x(α) (u1 ) = u1e
ξ0 + u1 v(α)
1 (u1 )e
Proof. Recall that the local zero set of f is of the form
e
x = u1e
ξ0 + u1 v1e
τ0 ,
with (u1 , v1 ) in the local zero set of f (1) . The part (i) is then obvious
since the local zero set of f (1) reduces to the origin.
184
Let us now prove (ii). The local zero set of f (1) is made up of two
C ∞ curves intersecting transversaly at the origin where their tangents
are the two distinct lines of solutions of the equation
D2 f (1) (0) · (u1 , v1 )2 = 0.
Note that the v1 -axis is never one of these tangents since D2 f (0) ·
(e
τ0 )2 , 0. As a result, the projection of either tangent onto the u1 -axis
and along the v1 -axis a linear isomorphism and it follow that the two C ∞
curves can be parameterized by u1 (cf. Figure 4.1 below)
5.4. Solution Through the Morse Lemma....
143
Figure 4.1:
More rigorously, each curve is of the form
t → (u1 (t), v1 (t)),
(4.14)
a C ∞ function around the origin verifying u1 (0) = v1 (0) = 0 and
d
(u1 , v1 )|t=0 , 0
dt
(cf.Chapter 2, Corollary 3.1). Thus, the nonzero vector ((du1 /dt)(0), 185
(dv1 /dt)(0)) is tangent to the curve at the origin and, as it is not collinear
with the v1 -axis, one has
du1
(0) , 0.
dt
From the Inverse mapping theorem, the relation u1 = u1 (t) is equivalent to saying that t = t(u1 ), a C ∞ function around the origin such that
t(0) = 0. Therefore, the curve (4.14) is as well parametrized by
u1 → (u1 , v1 (t(u1 )))
as stated above. We have then proved that the local zero set of f (1) is
made of two C ∞ curves (u1 , v(α)
1 (u1 )), α = 1, 2 and u1 ǫ(−ρ, ρ), ρ > 0
small enough. As the vectors (1, (dv(α)
1 /du1 )(0)), α = 1, 2 are tangent
5. Introduction to a Desingularization Process....
144
to a different one of the two curves at the origin (and hence are not
collinear) we deduce
dv(1)
1
du1
(0) ,
dv(2)
1
du1
(0).
Finally, the local zero set of f is made of the two C ∞ curves
τ0 , α = 1, 2.
u1 ǫ(−ρ, ρ) → e
x(α) (u1 ) = u1e
ξ0 + u1 v(α)
1 (u1 )e
Both are tangent to the characteristic Ξ at the origin since
de
x(α)
(0) = e
ξ0 , α = 1, 2,
du1
186
and they are distinct, since their second derivatives at the origin
dv(α)
d2 e
x(α)
1
(0)
=
(0),
du1
du21
differ.
Remark 4.3. We get an a posteriori proof of the fact that the sign of det
D2 f (1) (0) is independent of the choice of the space T because this sign
is related to the structure of the local zero set of f .
We shall noe describe a convenient way for finding the local zero set
of f from that of f (1) . It wil be repeatedly used in the applications of the
next section and we shall refer to it as the “quadrant method”.
Identify the sum Ξ ⊕ T with the product Re
ξ0 × Re
τ0 , the vector e
τ0
e
being chosen so that ξ0 ,e
τ0 is a direct system of coordinates. In the
(u1 , v1 )-plane as well as in the product Re
ξ0 × Re
τ0 , we shall distinguish
the usual four quadrants (I), (II), (III) and (IV).
5.4. Solution Through the Morse Lemma....
(II)
145
(I)
(I)
(II)
(III)
(III)
(IV)
(IV)
Clearly, the mapping
187
(u1 , v1 ) → u1e
ξ0 + u1 v1e
τ0 ,
maps the quadrants (I), · · · , (IV) of the (u1 , v1 )-plane into the quadrants
(I), · · · , (IV) of Re
ξ0 × Re
τ0 according to the rule
(I) → (I)
(III) → (II)
(II) → (III)
(IV) → (IV)
Thus, each half-branch in the local zero set of f (1) located in (I)
(resp. (II), (III), (IV)) is transformed into a half-branch in the local zero
set of f located in (I) (resp. (III), (II), (IV)). Figure 4.2 below features
some possible diagrams. The simplest example of a mapping f verifying
the conditions of this section is f (u, v) = u2 ± v4 (or v2 ± u4 or, with θ
any fixed real number, (v cos θ − u sin θ)2 ± (u cos θ + v sin θ)4 ).
1
2
1
2
2
1
2
1
2
1
2
1
Figure 4.2: (a) local zero set of f (1) .
5. Introduction to a Desingularization Process....
146
2
1
2
1
2 2
2
1
1
2
1
1
Figure 4.2: (b) local zero set of f
5.5 Iteration of the Process.
188
We shall now assume that the local zero set of the mapping f (1) cannot
be determined through either the Implict function theorem of the Morse
lemma, that is to say
D f (1) (0) = 0,
(5.1)
detD2 f (1) (0) = 0.
(5.2)
From (2.13), we know that (∂2 f (1) /∂v21 )(0) , 0. In particular,
D2 f (1) (0) , 0,
189
(5.3)
and the problem of finding the local set of f (1) is the same as considered in §2 with the mapping f . The mapping f (1) has a characteristic
Ξ1 which is a one-dimensional subspace of the (u1 , v1 )-plane; considering any complement T 1 of Ξ1 and choosing arbitrary nonzero elements
e
ξ1 and e
τ1 in Ξ1 and T 1 respectively,we can reduce the problem to finding the local zero set of a new mapping f (2) obtained by blowing-up
f (1) along Ξ1 . The mapping f (2) depends on the new variables (u2 , v2 )
through the relation





(2)
f (u2 , v2 ) = 



1 (1)
f (u2e
ξ1 + u2 v2e
τ1 ) for u2 , 0,
u22
1 2 2 (1)
τ1 )2 for u2 = 0.
2 v2 D f (0) · (e
5.5. Iteration of the Process.
147
An important observation is that the relation (∂2 f (1) /∂v21 )(0) , 0
keeps the characteristic Ξ1 from being the v1 -axis. This will be summarized by saying that the characteristic Ξ1 is not vertical. Such a notion
makes no sense as far as the characteristic Ξ is concerned, unless we
arbitrarily fix a system (u, v) of coordinates in the original plane R2 .
If D f (2) (0) , 0, the local zero set of f (2) can be found through the
Implicit function theorem. From Theorem 3.1, we deduce the local zero
set of f (1) and, finally, the local zero set of f . If D f (2) (0) = 0 and det
D2 f (2) (0) = 0, the mapping f (2) has a characteristic Ξ2 which is not
vertical in its ambient space (u2 , v2 ) and the problem reduces to finding
the local zero set of a third iterate f (3) depending on the new variables
(u3 , v3 ). More generally, we have
190
Theorem 5.1. Assume that the iterates f (1) , · · · , f (m) are defined and
D f (m) (0) , 0. After changing e
ξ0 into −e
ξ0 if necessary and setting Ξ+ =
R+ Ξ, the local zero set of f is made up of two continuous half-branches
emerging from the origin in the half-space Ξ+ ⊕ T . These half-branches
are of class C ∞ away from the origin and tangent to the characteristic
Ξ at the origin.
Proof. The mappings f (1) , · · · , f (m) are defined after the choice of non
zero elements e
ξ0 , · · · , e
ξm−1 and e
τ0 , · · · ,e
τm−1 in the characteristics Ξ,
Ξ1 , · · · Ξm−1 and some given complements T, T 1 , · · · , T m−1 . In the (u j ,
v j ) plane, e
ξ j and e
τ j are of the form e
ξ j = (α j , β j ) and e
τ j = (λ j , µ j ), with
α j , 0 since the characteristic Ξ j is not vertical for 1 ≤ j ≤ m − 1. Clearly, the iterate f (m) is the first iterate of f (m−1) . Thus, after
changing e
ξm−1 into −e
ξm−1 , if necessary (which does not affect the existence of the mth iterate f (m) or the condition D f (m) (0) , (0)), we know
from Theorem 3.1 that the local zero set of f (m−1) is made up of two
half-branches (ρ > 0 small enough, ψ an origin-presering C ∞ local diffeomorphism of R)
α√
e
0 ≤ um < ρ − e
x(α)
um )e
τm−1 , α = 1, 2.
m−1 (um ) = um ξm−1 + um ψ((−1)
In the system (um−1 , vm−1 ), this becomes
α√
um ),
u(α)
m−1 (um ) = αm−1 um + λm−1 um ψ((−1)
148
5. Introduction to a Desingularization Process....
α√
v(α)
um ).
m−1 (um ) = βm−1 um + µm−1 um ψ((−1)
191
(5.4)
(α)
1
The above mappings u(α)
m−1 (·) and vm−1 (·) are of class C in [0, ρ) with
(α)
/dum )(0) = αm−1 , 0.
(dum−1
(5.5)
Now, from §2, the local zero set of f (m−2) is made of elements of the
form
e
xm−2 = um−1e
ξm−2 + um−1 vm−1e
τm−2 ,
with (um−1 , vm−1 ) in the local zero set of f (m−1) . From (5.4), e
xm−2 =
(α)
e
xm−2 (um ), α = 1, 2, with
(α)
(α)
e
e
xm−2 (um ) = u(α)
τm−2 ,
m−1 (um )ξm−2 + um−1 (um )vm−1 (um )e
which, in the system (um−2 , vm−2 ), expresses as
(α)
(α)
(α)
u(α)
m−2 (um ) = αm−2 um−1 (um ) + λm−2 um−1 (um )vm−1 (um ),
(α)
(α)
(α)
v(α)
m−2 (um ) = βm−2 um−1 (um ) + µm−2 um−1 (um )vm−1 (um ).
(α)
1
The mappings u(α)
m−2 (·)vm−2 (·) are of class C in (0, ρ) and, from (5.5)
(du(α)
m−2 .dum )(0) = αm−2 αm−1 , 0, α = 1, 2,
since αm−2 , 0. Iterating the process, we find that the local zero set of
f is of the form
(α)
(α)
e
τ0 , α = 1, 2,
e
x(α) (um ) = u(α)
1 (um )ξ0 + u1 (um )v1 (um )e
(α)
1
where the functions u(α)
1 (·) and v1 (·) are of class C in [0, ρ), vanish as
the origin and
(du(α)
1 /dum )(0) = α1 · · · αm−1 , 0.
192
After shrinking ρ > 0 if necessary, the sign of u(α)
1 (um ) is that of the
product α1 · · · αm−1 for 0 < um < ρ and both values α = 1 and α = 2.
Hence our assertion, since all the mappings we have considered are C ∞
away from the origin.
5.5. Iteration of the Process.
149
When Theorem 5.1 applies, the partical determination of the local
zero set of f from that of f (m) follows by successive applications of the
“quadrant method” described in §4. Note, however, for 1 ≤ j ≤ m − 1,
that the quadrants (I), · · · , (IV) used for finding the local zero set of
f ( j) from the local zero set of f ( j+1) are (in general) different from the
quadrants (I), · · · , (IV) used for finding the local zero set of f ( j−1) from
that of f ( j) .
Figure 5.1 (a), (b), (c) below gioves an example of the local zero set
of f (2) , f (1) and f respectively when Theorem 5.1 applies with m = 2. A
mapping whose local zero set looks like that in Figure 5.1(c) (and for
which the process stops at m = 2 of course) is
f (u, v) = (v − u2 )2 + u5 .
(II)
(I)
(III)
(IV)
Figure 5.1: (a) local zero set of f (2) .
5. Introduction to a Desingularization Process....
150
(III)
(I)
(II)
(IV)
1
1
(II)
(III)
(I)
2
(IV)
2
Figure 5.1: (b) local zero set of f (1)
2
(II)
(I)
1
(III)
(IV)
Figure 5.1: (c) local zero set of f
193
194
Remark 5.1. In theory at least, applying Theorem 5.1 does not require
the computation of f (1) , · · · , f (n) and reduces to checking some conditions on the partial derivatives of f . For a general m, this follows
from the proof of the intrinsic character of the process, namely, that
whether or not it stops at some rank m with D f (m) (0) , 0 is independent
of the choice of the complements T, T 1 , · · · , T m−1 and of the elements
e
ξ0 , · · · , e
ξm−1 and e
τ0 , · · · ,e
τm−1 for defining the iterates f (1) , · · · , f (m)3 .
3
Observe that modifying T, · · · , T m−2 affects the characteristics Ξ1 , · · · , Ξm−1 as
well.
5.5. Iteration of the Process.
151
For m = 1, this was shown in § 3 (and the result is obvious). The proof
of this assertion for any m is very technical and will not be given in
these notes (cf. Rabier [32])4 . We shall only give the example of the
case when m = 2; The iterate f (2) is defined under the conditions (5.1)
and (5.2), namely (cf. §§3 and 4)
D3 f (0) · (e
ξ0 )3 = 0,
1 4
(D f (0) · (e
ξ)4 )(D2 f (0) · (e
τ0 )2 ) − (D3 f (0) · ((e
ξ0 )2 ,e
τ0 ))2 = 0.
3
(5.6)
(5.7)
Now, a nonzero element e
ξ1 of the characterisitc Ξ1 is
1
e
τ0 )2 ), − D3 f (0) · ((e
ξ0 )2 ,e
τ0 )).
ξ1 = (D2 f (0) · (e
2
From §3, the condition D f (2) (0) , 0 is equivalent to
D3 f (1) (0) · (e
ξ1 )3 , 0.
Using (5.7) and formula (2.12), this can be rewritten as
4 5
(D F(0) · (e
ξ0 )5 )(D2 f (0) · (e
τ0 )2 ) − 8(D4 f (0) · ((e
ξ0 )3 ,e
τ0 ))(D3 f (0)
5
· ((e
ξ0 )2 ,e
τ0 )) + 4D3 f (0) · (e
ξ0 , (e
τ0 )2 )(D4 f (0) · (e
ξ0 )4 ) , 0.
(5.8)
When (5.6) and (5.7) hold (which is independent of the choice of 195
e
ξ0 ǫΞ and e
τ0 from Proposition 4.1), it is easy to see that (5.8) also is
independent of e
ξ0 ǫΞ and e
τ0 . However, the reader can already guess that
proving the intrinsic character of the process by using formulas such as
(5.6)-(5.8) is impossible in the general case.
Assume now that D f (2) (0) = 0. Then, the local zero set of f (2)
can be found through the Morse lemma and Theorem 4.1 provides the
structure of the local zero set of f (1) , from which the structure of the
local zero set of f is easily derived. More generally, we have
4
However, § 6 gives a partial result in this direction.
152
5. Introduction to a Desingularization Process....
Theorem 5.2. Assume that the iterates f (1) , · · · , f (m) are defined and
D f (m) (0) = 0, det D2 f (m) (0) , 0. Then, the local zero set of f reduces
to the origin if det D2 f (m) (0) > 0 and is made up of exactly two distinct
curves of class C ∞ tangent to the characteristic Ξ at the origin if det
D2 f (m) (0) < 0.
Proof. It is possible to parallel the proof of Theorem 5.1. However, we
are going to give a more “geometrical” one. We limit ourselves to the
case m = 2, the general situation being identical by repeating the same
arguments. From §2 we know that the local zero set of f is of the form
196
e
x = u1e
ξ0 + u1 v1e
τ0 ,
with (u1 , v1 ) in the local zero set of f (1) . The assertion is then obvious if
det D2 f (2) (0) > 0 since the local zero set of f (1) reduces to the origin (cf.
Theorem 4.1). Assume then that det D2 f (2) (0) < 0 so that the local zero
set of f (1) is made of two C ∞ curves tangent to the characteristic Ξ1 at
the origin as Theorem 4.1 states. In addition, from the proof of Theorem
4.1, these curves coincide with the graphs of two C ∞ functions defined
around the origin in Ξ1 with values in T 1 . The important point is that
this property is not affected, if T 1 is replaced by any other complement
of T 1 (express the general form of a linear change of coordinates leaving the characteristic Ξ1 globally invariant and use the Implicit function
theorem). In particular, as the characteristic Ξ1 is not vertical, this complement can be taken as the v1 -axis R(0, 1). Thus, the local zero set of
f (1) coincides with the graphs of two C ∞ functions defined around the
origin Ξ1 with values in R(0, 1). Because the characterisitc Ξ1 is not
vertical again, its projection onto the u1 -axis is a linear isomorphism. It
follows that the local zero set of f (1) coincides with the graphs of two
C ∞ functions v(α)
1 (u1 ), α = 1, 2, defined for |u1 | < ρ(ρ > 0 small enough)
(α)
verifying v1 (0) = 0. Hence, the local zero set of f is made up of the
two curves
u1 ǫ(−ρ, ρ) → e
x(α) (u1 ) = u1e
ξ0 + u1 v(α)
τ0 , α = 1, 2.
1 (u1 )e
Theses two curves are tangent to the characteristic Ξ at the origin
and they are distinct, since they coincide with the graphs of the two
5.5. Iteration of the Process.
197
153
e
distinct functions u1 v(α)
τ0 . Finally, they are of class C ∞ ,
1 (u1 ) in Rξ0 × Re
(α)
since the functions v1 are and the proof is complete.
For practically finding the local zero set of f from the local zero set
of f (m) , the quadrant method can be applied again. We now show it on
the example when
f (u, v) = u2 + 2uv2 + u3 + 2u2 v + v4 + 2uv3 + uv4 + v5 + v6 .
Clearly, f (0) = 0, D f (0) = 0, D2 f (0) , 0 and det D2 f (0) = 0.
Here, the characteristic Ξ is the space Ξ = R(0, 1). It is convenient to
take T = R(1, 0) (however, the process being intrinsic, all the possible
choices are equivalent). If so, with e
ξ0 = (0, 1) and e
τ0 = (1, 0), we find
f (1) (u1 , v1 ) =
1
1
f (u1 (0, 1) + u1 v1 (1, 0)) = 2 f (u1 v1 , u1 ) =
2
u1
u1
= v21 + 2u1 v1 + u1 v31 + u1 v21 + u21 + 2u21 v1 + u31 v1 + u31 + u41 .
We see that D f (1) (0) = 0 while the quadratic part of f (1) is (u1 +v1 )2 .
Thus, the characteristic Ξ1 is Ξ1 = R(1, −1) and we choose T 1 = R(0, 1).
Taking e
ξ1 = (1, −1) and e
τ1 = (0, 1), we find
f (2) (u2 , v2 ) =
1
1 (1)
f (u2 (1, −1) + u2 v2 (0, 1)) = 2 f (1) (u2 , u2 (v2 − 1))
2
u2
u2
= v22 − u22 + 4u22 v2 + u2 v22 + u22 v32 − 3u22 v22 .
The mapping f (2) verifies D f (2) (0) = 0 and det D2 f (2) (0) = −4 < 0.
The two lines in the local zero set of D2 f (2) (0) · (u2 , v2 )2 are the lines 198
u2 = v2 and u2 = −v2 . Figure 5.2 showshow the quadrant method allows
to find the two curves in the local zero set of f .
5. Introduction to a Desingularization Process....
154
2
(II)
(I)
1
(IV)
(III)
2
1
(a) local zero set of f (2)
(II)
2
2
(I)
(I)
(II)
1
1
2
(III)
2
(III)
1
(IV)
1
(IV)
(b) local zero set of f (1)
2
1
1
2
2
1
2
1
(c) local zero set of f
Figure 5.2:
5.6. Partial Results on the Intrinsic Character of the Process
155
5.6 Partial Results on the Intrinsic Character of the
Process
A natural question we have already mentioned is to know whether the 199
process described in §5 is intrinsic or depends on the successive spaces
T, T 1 , · · · , T m−1 and elements e
ξ0 , e
ξ1 , · · · , e
ξm−1 and e
τ0 ,e
τ1 , · · · ,e
τm−1 cho(1)
sen for defining the iterates f , · · · , f (m) . In § 3, we saw that the condition D f (1) (0) , 0 was independent of the choice of T and e
ξ0 ,e
τ0 and
(1)
we also proved a similar result as concerns the conditions D f (0) = 0
and det D2 f (1) (0) , 0 together (cf. Proposition 4.1). The existence of a
second iterate f (2) (i.e. the condition D f (1) (0) = 0, det D2 f (1) (0) = 0)
is then independent of T and e
ξ0 ,e
τ0 . Given such a space T and elements
e
ξ0 ǫΞ,e
τ0 ǫT from which f (1) is defined, the existence of a third iterate
f (3) is then independent of the choice of the complement T 1 of the characteristic Ξ1 and of the non-zero elements e
ξ1 ǫΞ1 , e
τ1 ǫT 1 . But whether or
not its existence is independent of the initial choice of T and e
ξ0 ,e
τ0 is not
ensured by our previous results. However, we can reduce the question as
follows. Since the characteristic Ξ1 is not vertical and since the result is
known to be independent of T 1 , it suffices to prove that the existence of
f (3) is independent of T, e
ξ0 ,e
τ0 when T 1 is taken as the v1 -axis. Suppose
then that this step has been solved successfully. We deduce that the existence of a fourth iterate f (4) is independent of the choice of T 1 , T 2 and 200
of the non-zero elements e
ξ1 , e
ξ2 and τ̃1 , τ̃2 in the characterisitcs Ξ1 , Ξ2
and their complements T 1 , T 2 respectively. Again, the non-dependence
on the initial choice of T and e
ξ0 ,e
τ0 is not a consequence of the results
we have established so far, but the problem can be reduced to the case
when T 1 is taken as the v1 -axis and T 2 is taken as the v2 -axis. More
generally, we see that proving that the process of §5 is intrinsic reduces
to proving for every m that it is independent of T and e
ξ0 ,e
τ0 when the
space T 1 , · · · , T m−1 are taken as the v1 −, · · · , vm−1 -axes.
Therefore, we shall henceforth assume that the sequence f (1) , . . . ,
(n)
f is defined when T j is taken as the v j -axis, 1 ≤ j ≤ m − 1. Equivalently, we shall say that each space T j , 1 ≤ j ≤ m − 1 is vertical. A proof
of the independence of T based on some generalization of Proposition
4.1 is not available. Indeed, Proposition 4.1 allows to prove the first in-
5. Introduction to a Desingularization Process....
156
201
dependence result and is based on “linear” arguments while the question
of the independence of T in the general case is a higher order (hence)
nonlinear problem. By showing that the two half-branches/curves (following that either Theorem 5.1 or Theorem 5.2 applies) in the local zero
set of f have a contact of order m exactly at the origin, it is possible to
establish that the process is intrinsic if, for choice of T , D f (m) (0) , 0 or
D f (m) (0) = 0 and det D2 f (m) (0) < 0 (thus, in such a case, m and either
condition D f (m) (0) , 0 or D f (m) (0) = 0 and det D2 f (m) (0) < 0 is independent of T ). If D f (m) (0) = 0 and det D2 f (m) (0) > 0, the local zero
set of f reduces to the origin and no useful information of an invariant
geometrical character can be derived from the structure of the local zero
set of f . Neverthless, this situation also can be shown to be intrinsic.
The details of the above assertions can be found in Rabier [32]. Here, it
will be sufficient for our purposes to prove that the process is intrinsic
in a particular case that we shall now describe.
We begin with a definition. We shall say that the characteristic Ξ j
is horizontal if it coincides with u j -axis in its ambient space (u j , v j ).
Similarly, after the choice of a system of coordinates (u, v) in the original plane R2 , we shall say that the characteristic Ξ is horizontal if it
coincides with the u-axis. The v-axis will be referred to as the vertical
axis.
Remark 6.1. Note that we can always make a choice of the system (u, v)
so that the characteristic Ξ is horizontal but we have no freedom of making the characteristics Ξ1 , · · · , Ξm−1 horizontal. Conversely, the characteristics Ξ1 , · · · , Ξm−1 being horizontal is independent of the system
(u, v) of coordinates in the original plane R2 , since no particular system
is involved in their definition.
Theorem 6.1. Let (u, v) be a system of coordinates in which the characteristic Ξ is not vertical. Then, the characteristics Ξ, Ξ1 , · · · , Ξm−1 are
horizontal if and only if
(∂
202
(∂ j f /∂ui )(0) = 0, 0 ≤ j ≤ 2m,
(6.1)
f /∂u ∂v)(0) = 0, 0 ≤ j ≤ m.
(6.2)
j+1
j
5.6. Partial Results on the Intrinsic Character of the Process
157
Proof. Note that the relations (6.1) and (6.2) are independent of the
change of coordinates leaving the u-axis invariant. Indeed, such a
change of coordinates is of the form
U = au + bv,
V = cv,
with ac , 0. Thus, at the origin
∂ j /∂u j = a j ∂ j /∂U j , ∂ j+1 /∂u j ∂u = a j b∂ j+1 /∂U j+1 + a j c∂ j+1 /∂U j ∂V,
(6.3)
so that the relations (6.1) and (6.2) hold in the system (U, V) if and only
if they hold in the system (u, v).
As the characteristic Ξ is not vertical, it is therefore not restrictive to
suppose that T is the v-axis (by changing the latter, not T ). On the other
hand, the equivalence is true with m = 1 as is immediately checked.
Assume then m > 1 and the equivalence holds. We shall prove that
the characteristics Ξ, Ξ1 , · · · , Ξm exist and are horizontal if and only if
(6.1) and (6.2) hold with m + 1 replacing m. Let then the characteristics
Ξ, Ξ1 , · · · , Ξm−1 exist and be horizontal. In particular, this is true with
Ξ, Ξ1 , · · · , Ξm−1 , which is equivalent to (6.1) - (6.2) by hypothesis. Now,
saying that Ξm exists and is horizontal means that
D f (m) (0) = D2 f (m) (0) · (1, 0) = 0ǫL (R2 , R),
a condition expressed by the four scalar equations
(∂ f (m) /∂um )(0) = (∂ f (m) /∂vm )(0)
= (∂2 f (m) /∂um ∂vm )(0)
= (∂2 f (m) /∂u2m )(0) = 0.
Applying formula (2.12) with f (m) , f (m−1) , e
ξm−1 and e
τm−1 replac- 203
(1)
e
e
ing f , f, ξ0 and e
τ0 respectively, where ξm−1 and e
τm−1 are two given
nonzero elements of Ξm−1 and T m−1 (= vm−1 -axis), the equation
(∂ f (m) /∂vm )(0) = 0 is automatically satisfied. The remaining three ones
are
D3 f (m−1) (0) · (e
ξm−1 )3 = D4 f (m−1) (0) · (e
ξm−1 )4
158
5. Introduction to a Desingularization Process....
= D3 f (m−1) (0) · ((e
ξm−1 )2 ,e
τm−1 ) = 0
By hypothesis, e
ξm−1 and e
τm−1 ae collinear with the um−1 - and vm−1 axes respectively. Therefore, we get the equivalent relation
(∂3 f (m−1) /∂u3m−1 )(0) = (∂4 f (m−1) /∂u4m−1 )(0)
= (∂3 f (m−1) /∂u2m−1 ∂vm−1 )(0) = 0.
Again, with the same arguments, each of these equations can be expressed in terms of f (m−2) and the variables um−2 and vm−2 . Iterating the
process and since Ξ and T coincide with the u- and v- axis respectively,
we finally find
(∂2m+1 f /∂u2m+1 )(0) = (∂2m+2 f /∂u2m+2 )(0) = (∂m+2 f /∂um+1 ∂v)(0) = 0,
to be necessary and sufficient condition for the characteristic Ξm to exist and be horizontal, under the equivalent assumptions that Ξ, Ξ1 , · · · ,
Ξm−1 exist and are horizontal or that (6.1)-(6.2) hold. The equivalence
at rank m + 1 follows.
Remark 6.2. Theorem 6.1 is clearly independent of the choice of T . By
the arguments of the beginning of this section, the fact that Ξ1 , · · · , Ξm−1
are horizontal is then independent of any choice T, T 1 , · · · , T m−2 for
defining the iterates f (1) , · · · , f (m−1) .
204
Remark 6.3. (Intrinsic character of the process when the characteristics Ξ1 , · · · , Ξm−1 are horizontal): From Theorem 6.1 and by successive
applications of formula (2.12), it is easily seen in a system (u, v) of cordinates in which the characteristic Ξ is horizontal and its complement T
vertical (so that e
ξ0 = (α0 , 0), e
τ0 = (0, µ0 )) and when the characteristics
Ξ1 , · · · , Ξm−1 are horizontal, that, setting e
ξ j = (α j , 0) and e
τ j = (0, µ j )
(∂ f (m) /∂um )(0) = [1/(2m + 1)!]α3m−1 · · · α02m+1
(∂2m+1 f /∂µ2m+1 )(0),
(6.4)
(∂2 f (m) /∂um ∂vm )(0) = [1/(m + 1)!]µm−1 · · · µ0 α2m−1 · · · αm+1
0
(∂m+2 f /∂um+1 ∂v)(0),
(6.5)
5.6. Partial Results on the Intrinsic Character of the Process
159
(∂2 f (m) /∂u2m )(0) = [2/(2m + 2)!]α4m−1 · · · α02m+2 (∂2m+2 f /∂u2m+2 )(0),
(6.6)
(∂2 f (m) /∂v2m )(0) = µ2m−1 · · · µ20 (∂2 f /∂v2 )(0).
(6.7)
Together with the relation (∂ f (m) /∂vm )(0) = 0 (cf. Theorem 6.1),
these relations show that whether or not the characteristic Ξm exist (but
is not necessarily horizontal) is independent of T . Indeed, the conditions
D f (m) (0) , 0 or D f (m) (0) = 0 and det D2 f (m) (0) , 0 are independent of
the change of coordinates in the (u, v)-plane leaving the u-axis invariant
because, in such a change U = au + bv, V = cv(ac , 0), the right
hand sides of (6.4)-(6.7), are multiplied by a−2m−1 , a−m−1 c−1 , a−2m−2
and c−2 respectively (cf. (6.3)). Using the arguments at the beginning of this section and Remark 6.2, we conclude that the assumption: “Ξ1 , · · · , Ξm−1 exist and are horizontal and Ξm does not exist”
is independent of any choice of T, T 1 , · · · , T m−1 (and, of course, of
e
ξ0 , e
ξ1 , · · · , e
ξm−1 and e
τ0 ,e
τ1 , · · · ,e
τm−1 ) for defining f (1) , · · · , f (m) .
In other words, the rank m at which the process stops (if at all) is 205
intrinsically linked to f .
Corollary 6.1. Let f verify the condition
f (u, 0) = 0
(6.8)
for u around the origin. Then, the characteristic Ξ is horizontal and
Ξ1 , · · · , Ξm−1 exist if and only if
(∂ j+1 f /∂u j ∂v)(0) = 0, 0 ≤ j ≤ m.
(6.9)
If so, they are horizontal, Finally, if there is a largest integer m (≥ 1
necessarily) such that (6.9) holds, the process §5 ends at the step m
exactly and the local zero set of f is the union of the u-axis and one
distinct C ∞ curve tangent to the u-axis at the origin.
Proof. Under the assumption (6.8), the characteristic Ξ is horizontal.
Indeed, one has (∂2 f /∂u2 )(0) = 0 and, since det D2 f (0) = 0, we deduce
(∂2 f /∂u∂v)(0) = 0. Thus, D2 f (0)·(1, 0) = 0ǫL (R2 , R), a relation which
expresses that Ξ is horizontal. Taking e
ξ0 = (1, 0) and e
τ0 = (0, 1) (we
160
5. Introduction to a Desingularization Process....
leave it to the reader to check that any other choice leads to the same
result) the iterate f (1) is


1


 u21 f (u1 , u1 v1 ) for u1 , 0,
(1)
f (u1 , v1 ) = 


 21 v21 (∂2 f /∂v2 )(0) for u1 = 0.
206
Clearly, from (6.8), f (1) (u1 , 0) = 0 for u1 around the origin. Hence,
if Ξ1 exists, it must be horizontal for the same reason as Ξ is. More
generally, each iterate f ( j) verifies f ( j) (u j , 0) = 0 for u j around the origin
and Ξ j must be horizontal if it exists. Theorem 6.1 is then available and
states that assuming that Ξ1 , · · · , Ξm−1 exist is equivalent to assuming
that the relations (6.1) and (6.2) hold. But (6.1) is automatically satisfied
because of (6.8) and the first part of our assertion follows.
Let now m be the largest integer such that (6.9) holds (if at all). Then
(∂ j+1 f /∂u j ∂v)(0) = 0, 0 ≤ j ≤ m,
(∂m+2 f /∂um+1 ∂v)(0) , 0,
and the characteristics Ξ1 , · · · , Ξm−1 exist (m ≥ 1) while Ξm does not.
This means either D f (m) (0) , 0 or D f (m) (0) = 0 and det D2 f (m) (0) , 0.
From Theorem 5.1 and 5.2, it is immediately checked that the only case
compatible with the fact that the trivial branch (u, 0) for |u| small enough
is in the local zero set of f is when D f (m) (0) = 0, det D2 f (m) (0) < 0 and
the conclusion is given by Theorem 5.2.
Remark 6.4. The second part of Corollary 6.1 can also be derived from
formulas (6.4)-(6.7) and Theorem 5.2.
207
Remark 6.5. The mapping f verifying the condition f (u, 0) = 0 for
u around the origin, it can of course happen that all the derivatives
(∂ j+1 f /∂u j ∂v)(0) vanish. As all the derivatives (∂ j f /∂u j )(0) also vanish
because of the special form of f , we deduce, when f is real-analytic,
that f can be written as f (u, v) = v2 h(u, v). Since D2 f (0) , 0 by hypothesis, one has h(0) , 0 and the local zero set of f reduces to the
trivial branch. If f is not real-analytic, the result is false as is easily seen
2
2
by taking the example of f (u, v) = v2 − ve−1/(u +v ) . This observation
will be generalized in §8.
5.7. An Analytic Proof of Krasnoselskii’s Theorem....
161
5.7 An Analytic Proof of Krasnoselskii’s Theorem
in a Non-Classical Particular Case.
In this section, we consider again the problem of finding the local zero
set of a mapping of the form
G(µ, x) = (I − (λ0 + µ)L)x + Γ(µ, x),
(7.1)
defined on a neighbourhood of the origin in the product R× X and taking
its values in the real Banach space X. As in Chapter 1, the operator
LǫL (X) is supposed to be compact and the nonlinear operator Γ verifies
Γ(µ, 0) = 0,
(7.2)
D x Γ(µ, 0) = 0,
(7.3)
for |µ| small enough. For the sake of simplicity, we shall assume that Γ
is of class C ∞ , although this hypothesis of regularity can be weakened
without affecting the final results.
Recall the notation introduced in Chapter 1
X1 = Ker(I − λ0 L),
Y2 = Range(I − λ0 L)
(7.4)
(7.5)
while X2 and Y1 are two topological caomplements of X1 and Y2 re- 208
spectively. Denoting by Q1 and Q2 the projection operators onto the
spaces Y1 and Y2 , the Lyapunov-Schmidt reduction leads to the reduced
equation (cf. Chapter 1, §3)
f (µ, x) = −
µ
µ
Q1 x − Q1 ϕ(µ, x) + Q1 Γ(µ, x + ϕ(µ, x)) = 0,
λ0
λ0
(7.6)
for (µ, x) around the origin of R× X1 , where the mapping ϕ (here of class
C ∞ ) is characterized by
−
µ
Q2 x+(I−λ0 L)ϕ(µ, x)−µQ2 Lϕ(µ, x)+Q2 Γ(µ, x+ϕ(µ, x)) = 0, (7.7)
λ0
and verifies the conditions
ϕ(µ, 0) = 0
(7.8)
162
5. Introduction to a Desingularization Process....
for |µ| small enough and
Dϕ(0) = 0.
(7.9)
When the algebraic and geometric multiplicities of λ0 coincide,
namely
X = X1 ⊕ Y2 (= Ker(I − λ0 L) ⊕ Range(I − λ0 L)),
(7.10)
the problem was studied in Chapter 1 if dim Ker (I − λ0 L) = 1 and in
Chapter 3 if dim Ker (I − λ0 L) = n ≥ 2. In what follows, we shall
assume
dim Ker(I − λ0 L) = 1,
(7.11)
209
but drop the condition (7.10). This means that we consider the case
X1 ⊂ Y2 (i.e. Ker (I − λ0 L) ⊂ Range (I − λ0 L)).
(7.12)
Thus, for xǫX1 , Q1 x = 0 and Q2 x = x so that the reduced mapping
(7.6) becomes
µ
f (µ, x) = − Q1 ϕ(µ, x) + Q1 Γ(µ, x + ϕ(µ, x)),
(7.13)
λ0
while the characterization (7.7) of the mapping ϕ can be rewritten as
−
µ
x + (I − λ0 l)ϕ(µ, x) − µQ2 Lϕ(µ, x) + Q2 Γ(µ, x + ϕ(µ, x)) = 0. (7.14)
λ0
The generalized null-space of the operator (I − λ0 L) will play a key
role. Recall that it is defined as
Ker(I − λ0 L)γ ,
where γ is the smallest positive integer such that 5
′
Ker(I − λ0 L)γ = Ker(I − λ0 L)γ ,
for every γ′ ≥ γ. Because of (7.12), one has γ ≥ 2 and, by definition the
algebraic multiplicity of λ0 is the integer
dim Ker(I − λ0 L)γ .
5
The existence of γ is known from the spectral theory of compact operators as recalled in Cahpter 1.
5.7. An Analytic Proof of Krasnoselskii’s Theorem....
163
Remark 7.1. As dim Ker (I − λ0 L) = 1 by hypothesis, on has
dim Ker(I − λ0 L)γ = γ.
(7.15)
Indeed, by a simple induction argument
dimKer(I − λ0 L) j ≤ j,
(7.16)
for every jǫN and
210
dim ker(I − λ0 L) j−1 < dim Ker(I − λ0 L) j ,
for j ≤ γ, by definition of γ. Such a simple relation as (7.15) between γ
and dim Ker (I−λ0 L)γ is no longer available when dim Ker (I−λ0 L) ≥ 2.
Applying Krasnoselskii’s theorem (Theorem 1.2) of Chapter 1 it follows that existence of nontrivial solutions ot the equation G(µ, x) = 0 arbitrarily close to the origin of R × X is ensured when γ is odd. We shall
complement this result in a particular case by showing that bifurcation
occurs regardless of the parity of γ and obtain a precise description of
the local zero set of G. Before that, we need to establish some preliminary properties of the reduced mapping f in (7.13). First, let e0 denote
a given eigenvector of L associated with the characteristic value λ0 so
that every xǫX1 = Ker(I − λ0 L) can be written in the form x = ǫe0 ,
ǫ ∈ R. With an obvious abuse of notation, the reduced mapping f in
(7.13) identifies with
f (µ, ǫ) = −
µ
Q1 ϕ(µ, ǫ) + Q1 Γ(µ, ǫe0 + ϕ(µ, ǫ)),
λ0
(7.17)
where ϕ(µ, ǫ) is characterized by (cf. (7.14))
−
µǫ
e0 +(I−λ0 L)ϕ(µ, ǫ)−µQ2 Lϕ(µ, ǫ)+Q2 γ(µ, ǫe0 +ϕ(µ, ǫ)) = 0 (7.18)
λ0
Differentiating (7.17) with respect to ǫ, we find
µ
∂ϕ
∂ϕ
∂f
(µ, ǫ) = − Q1 (µ, ǫ) + Q1 D x Γ(µ, ǫe0 + ϕ(µ, ǫ)) · (e0 +
(µ, ǫ)).
∂ǫ
λ0 ∂ǫ
∂ǫ
(7.19)
5. Introduction to a Desingularization Process....
164
For ǫ = 0 and from (7.3) and (7.8), this expression is simply
∂f
µ
∂ϕ
(µ, 0) = − Q1 (µ, 0).
∂ǫ
λ0 ∂ǫ
211
(7.20)
In particular
∂f
(0) = 0.
(7.21)
∂ǫ
Next, differentaiting (7.19) with respect to ǫ at the origin, we get
∂2 f
(0) = Q1 D2x Γ(0) · (e0 )2 .
∂ǫ 2
(7.22)
Finally, differentiating (7.20) with respect to µ for any order j at the
origin, it is easy to see that
j!
∂ j+1 f
∂ jϕ
(0)
=
−
(0).
Q
1
λ0 ∂µ j−1 ∂ǫ
∂µ j∂ǫ
(7.23)
Theorem 7.1. Assume that
Q1 D2x Γ(0) · (e0 )2 ,
(7.24)
Then, the local zero set of the mapping G (7.1) is the union of the
trivial branch and exactly one distinct C ∞ curve tangent to the trivial
branch at the origin.
Proof. It suffices to prove an analogous result for the reduced mapping
f . As f (µ, 0) = 0, one has
∂2 f
∂f
(0) = 2 (0) = 0.
∂µ
∂µ
(7.25)
Meanwhile, for j = 1 in (7.23) and due to (7.9),
∂2 f
(0) = 0.
∂µ∂ǫ
(7.26)
5.7. An Analytic Proof of Krasnoselskii’s Theorem....
212
165
Thus, the relations (7.21)-(7.26) show that D f (0) = 0, det D2 f (0) =
0 and D2 f (0) , 0. According to Corollary 6.1, it suffices to prove that
there exists an index j (necessarily ≥ 2) such that (∂ j+1 f /∂µ j ∂ǫ)(0) , 0.
From (7.23), this is equivalent to showing that
Q1
∂ jϕ
(0) , 0.
∂µ j−1 ∂ǫ
for some j ≥ 2. To does this, we go back to the characterization of ϕ
(7.18). Differentiating this identify with respect to ǫ and setting ǫ = 0,
it follows from (7.3) that
!
∂ϕ
∂ϕ
µ
− µQ2 L
(µ, 0) = 0.
− e0 + (I − λ0 L)
λ0
∂ǫ
∂ǫ
Now, differentiating with respect to µ yields
∂2 ϕ
∂2 ϕ
∂ϕ
1
e0 + (I − λ0 L)
(µ, 0) − µQ2 L
(µ, 0) − Q2 L (µ, 0) = 0,
λ0
∂µ∂ǫ
∂µ∂ǫ
∂ǫ
(7.27)
which, because of (7.9), provides
−
(I − λ0 L)
∂2 ϕ
1
(0) = e0 .
∂µ∂ǫ
λ0
2
∂ ϕ
Assume first that γ = 2. If Q1 ∂µ∂ǫ
(0) = 0, one has
Range(I − λ0 L). Thus, there exists ξ ∈ X such that
(I − λ0 L)ξ =
(7.28)
∂2 ϕ
∂µ∂ǫ (0)
∈ Y2 =
∂2 ϕ
(0).
∂µ∂ǫ
(7.29)
1
,
λ0
(7.30)
Hence, from (7.28)
(I − λ0 L)2 ξ =
and consequently (I − λ0 L)3 ξ = 0. But, since γ = 2,
Ker(I − λ0 L)3 = Ker(I − λ0 L)2 .
5. Introduction to a Desingularization Process....
166
We deduce that (I − λ0 l)2 ξ = 0, which contradicts (7.30). Our as- 213
sertion is then proved when γ = 2. When γ ≥ 3, we shall use the same
method but we need some preliminary observations. First, by differentiating (7.27) at any order j − 1, j ≥ 2, at the origin, we get
(I − λ0 L)
∂ jϕ
∂ j−1 ϕ
(0)
=
(
j
−
1)Q
L
(0).
2
∂µ j−1 ∂ǫ
∂µ j−2 ∂ǫ
(7.31)
j−1
If there is 3 ≤ j ≤ γ such that Q1 ∂µ∂ j−2ϕ∂ǫ (0) , 0, the problem is
solved. Assume then
Q1
∂ j−1 ϕ
(0) = 0, 3 ≤ j ≤ γ,
∂µ j−2 ∂ǫ
or equivalently,
∂ j−1 ϕ
(0) ∈ Y2 = Range(I − λ0 L), 3 ≤ j ≤ γ
∂µ j−2 ∂ǫ
Clearly, the space Range (I − λ0 L) is stable under L and, for the
indices 3 ≤ j ≤ γ, (7.31) reads
(I − λ0 L)
∂ j−1 ϕ
∂ jϕ
(0)
=
(
j
−
1)L
(0).
∂µ j−1 ∂ǫ
∂µ j−2 ∂ǫ
In particular, for j = γ and applying (I − λ0 L) to both sides
(I − λ0 L)2
∂γϕ
∂γ−1 ϕ
(0)
=
(γ
−
1)L(I
−
λ
L)
(0).
0
∂µγ−1 ∂ǫ
∂µγ−2 ∂ǫ
If γ = 3, it follows from (7.28) that
(I − λo L)2
214
2
∂3 ϕ
(0) = 2 e0 ,
∂µ2 ∂ǫ
λ0
while, if γ ≥ 4, (7.32) provides
(I − λ0 L)2
γ−2 ϕ
∂γ ϕ
2 ∂
(0)
=
(γ
−
1)(γ
−
2)L
(0).
∂µγ−1 ∂ǫ
∂µγ−3 ∂ǫ
(7.32)
5.7. An Analytic Proof of Krasnoselskii’s Theorem....
167
Iterating the process, it is clear in any case that
(I − λ0 L)γ−1
(γ − 1)!
∂γ ϕ
(0) =
e0 .
γ−1
∂µγ−1 ∂ǫ
λ0
(7.33)
γ
As Q1 ∂µ∂γ−1ϕ∂ǫ (0) = 0, there is ξ ∈ X such that
(I − λ0 L)ξ =
∂γ ϕ
(0).
∂µγ−1 ∂ǫ
From (7.33)
(I − λ0 L)γ ξ =
(γ − 1)!
λγ−1
0
e0 .
(7.34)
Hence, (I − λ0 L)γ+1 ξ = 0. But, by the definition of γ, one has
Ker(I − λ0 L)γ+1 = Ker(I − λ0 L)γ . It follows that (I − λ0 L)γ ξ = 0, which
contradicts (7.34).
Remark 7.2. The theorem can be complemented by showing that
Q1
∂ jϕ
(0) = 0, 2 ≤ j ≤ γ − 1.
∂µ j−1 ∂ǫ
Indeed, let us denote by p(≥ 2) the first integer such that
Q1 (∂ p ϕ/∂µ p−1 ∂ǫ)(0) , 0.
From the proof of Theorem 7.1, we have p ≤ γ. With the arguments we
used for proving (7.33), we find
(I − λ0 L) j−1
∂ jϕ
( j − 1)!
(0) =
e0 , 2 ≤ j ≤ p.
j−1
j−1
∂µ ∂ǫ
λ0
Setting
215
e j−1
∂ jϕ
(0), 2 ≤ j ≤ p,
=
∂µ j−1 ∂ǫ
this relation yields
(I − λ0 L) j e j =
j!
e ,1
j 0
λ0
≤ j ≤ p − 1.
(7.35)
168
5. Introduction to a Desingularization Process....
Clearly, for 1 ≤ j ≤ p − 1, the vectors e0 , · · · , e j belong to the space
Ker(I − λ0 L) j+1 and they are linearly independent: indeed, assuming
j
X
α j eℓ = 0,
ℓ=0
and applying (I − λ0 L) j , · · · , (I − λ0 L) successively, it follows that α j =
· · · = α0 = 0. Owing to (7.16), we deduce that the vectors e0 , · · · , e j
form a basis of Ker (I − λ0 L) j+1 for 1 ≤ j ≤ p − 1. As a result,
Ker(I − λ0 L) p+1 = Ker(I − λ0 L) p .
(7.36)
To see this, consider an element ξǫKer(I − λ0 L) p+1 . Then, (I −
λ0 L)ξǫKer(I − λ0 L) p , namely
(I − λ0 L)ξ =
p−1
X
αℓ eℓ .
ℓ=0
By definition of p, one has e p−1 < Y2 = Range(I − λ0 L) (cf. (7.35))
while eℓ ǫRange(I − λ0 L), 0 ≤ ℓ ≤ p − 1. As (I − λ0 L)ξǫRange(I − λ0 L),
the coefficient α p−1 must vanish. But, if so,
(I − λ0 L)ξ =
216
p−2
X
ℓ=0
αℓ eℓ ǫKer(I − λ0 L) p−1 .
L) p
Hence, ξǫKer(I − λ0
and (7.36) follows. We conclude that p ≥ γ
and therefore p = γ.
This conclusion is important because, according to our discussion
of § 6, it means that the process of desingularization of the reduced
mapping f ends at the step γ − 1 exactly (i.e. the iterates f (1) , · · · , f (γ−1)
are defined and the local zero set of f (γ−1) is found through the Morse
lemma).
Remark 7.3. Theorem 7.1 holds as soon as we dan apply the Morse
lemma to the iterate f (γ−1) . According to the results of Chapter 2, this
requires f (γ−1) ∈ C 2 , which is the case if f ∈ C 2γ and hence if Γ ∈ C 2γ .
Also, the condition (7.3) can be weakened and replaced by
j
Dµ D x Γ(0) = 0, 1 ≤ j ≤ γ.
5.7. An Analytic Proof of Krasnoselskii’s Theorem....
169
Remark 7.4. As mentioned in §6, since the desingularization process
stops at rank γ − 1 exactly (cf. Remark 7.2), the two curves in the local
zero set of f (so those in the zero set of G) have a contact of order
γ − 1 exactly at the origin. Following the parity of γ and apart from the
symmetry with respect to the µ-axis, the local zero set of f is given by
Figure 7.1 below. This can be easily confirmed by using the “quadrant
method” described in § 4.
even
odd
Figure 7.1: Structure of the local zero set of f (7.17).
We shall complete
this section with an example: Let us take X = R2 217
!
1 1
and L =
so that λ0 = 1 and X1 = Y2 = R(1, 0) and we can choose
0 1
e0 = (1, 0) and X2 = Y1 = R(0, 1). With
Γ(µ, x) = Γ(x) = Γ(ǫ, w) = (w2 , ǫ 2 ),
it is immediate that the assumptions (7.2), (7.3) and (7.24) are satisfied
and γ = 2. The equation G(µ, x) = G(µ, ǫ, w) = 0 is
− µǫ − (1 + µ)w + w2 = 0,
− µw + ǫ 2 = 0.
The first equation yields
(7.37)
218
170
5. Introduction to a Desingularization Process....
√
(1 + µ) − ((1 + µ)2 + 4µǫ)
w = ϕ(µ, ǫ) =
2
2µǫ
=−
,
√
(1 + µ) + ((1 + µ)2 + 4µǫ)
so that
ϕ(µ, ǫ)e
−µǫ,
around the origin. By referring to the second equation (7.37), we find
the reduced mapping f (µ, ǫ) ∼ −µ2 ǫ + ǫ 2 . The solutions of the equation
−µ2 ǫ + ǫ 2 = 0 are (µ, 0) and (µ, µ2 ) and the local zero set of f is as in
Figure 7.1 with γ even. Note that the “natural” choice
Γ(ǫ, v) = (ǫ 2 , v2 )
cannot be treated here because the condition (7.24) fails (and hence
D2 f (0) = 0). Actually, the associated reduced equation is
−µϕ(µ, ǫ) + ϕ2 (µ, ǫ) = 0,
with
ϕ(µ, ǫ) =
219
(7.38)
ǫ(ǫ − µ)
1+µ
It is easily checked that the local zero set of the reduced mapping
is made up of the ithree transversal curves (µ, 0), (µ, µ) and (2ǫ/ [1 + ǫ+
√
((1 + ǫ)2 + 4ǫ 2 ) , ǫ) ∼ (ǫ 2 , ǫ). Observe in this case that Theorem 4.1
of Chapter 2 applies to the reduced mapping (7.38) because its first
nonzero derivative at the origin is of order 3 and the associated polynomial mapping verifies the condition (R − N.D.). This is not in contradiction with the analysis of Chapter 3, because the condition (1.7) (as
well as its generalized form (1.26)) of Chapter 3 fails.
5.8 The Case of an Infinite Process.
To be complete, we analyze now the case when the process of §5 is
endless. We begin with the following definition
5.8. The Case of an Infinite Process.
171
Definition 8.1. We shall say that the mapping f is m-degenerate if the
process of §5 stops after m steps exactly. When f is m-degenerate for
some m, f will be refered to as finitely degenerate. Finally, if the process
of § 5 is endless, we shall say that f is indefinitely degenerate (in short
∞-degenerate).
Remark 8.1. This definition assumes the intrinsic character of the process of §5, proved in [32], that we shall then admit, for Definition 8.1 to
make sense.
With this vocabulary, the study we made in §5 was devoted to the determination of the local zero set of a finitely degenerate mapping. When
f is ∞-degenerate, the structure of its local zero set can still be determined in a general framework. To see this, we need
Proposition 8.1. Assume that the mapping f is of the form
f (e
x) = (ρ(e
x))2 h(e
x),
(8.1)
where ρ and h are two C ∞ real-valued mappings and ρ(0) = 0. Then,
f verifies the condition D f (0) = 0, det D2 f (0) = 0, D2 f (0) , 0 if and 220
only if h(0) , 0 and Dρ (0) , 0. If so, f is ∞-degenerate.
Proof. By differentiating (8.1) and since ρ(0) = 0, we find D f (0) = 0
and D2 f (0) · (e
ξ, e
ζ) = 2h(0)(Dρ(0) · e
ξ)(Dρ(0) · e
ζ). Hence, D2 f (0) , 0
if and only if h(0) , 0 and Dρ(0) , 0. If so, the characteristic Ξ exists
with
Ξ = Ker Dρ(0)
(8.2)
Let T be a complement of Ξ in R2 and take two nonzero elements
e
ξ0 ǫΞ and e
τ0 ǫT : The mapping f (1) is defined by


1

ξ0 + u1 v1e
τ0 ))2 h(u1e
ξ0 + u1 v1e
τ0 ) for u1 , 0

 u21 (ρ(u − 1e
(1)
f (u1 , v1 ) = 


 h(0)(Dρ(0) · e
τ0 )2 v21 for u1 = 0.
Let us introduce the two C ∞ real - valued mappings
Z 1
ρ(1) (u1 , v1 ) =
Dρ(su1e
ξ0 + su1 v1e
τ0 ) · (e
ξ0 + v1e
τ0 )ds,
0
5. Introduction to a Desingularization Process....
172
h(1) (u1 , v1 ) = h(u1e
ξ0 + u1 v1e
τ0 ).
One has ρ(1) (0) = 0, h(1) (0) = 0 and the derivative Dρ(1) (0) is the
nonzero linear mapping
1
(u1 , v1 )ǫR2 → (D2 ρ(0) · (e
ξ0 )2 )u1 + (Dρ(0) · e
τ0 )v1 .
2
R1
x) · e
xds, it is immediate that f (1) =
From the relation ρ(e
x) = 0 Dρ(se
ρ(1) h(1) with ρ(1) and h(1) verifying the same hypotheses as ρ and h.
Hence the existence of a second iterate f (2) and, more generally, of f (m) ,
m ≥ 1, so that f is ∞-degenerate.
The above result motivates the following definition.
221
Definition 8.2. We shall say that the ∞-degenerate mapping f is reducible if we can write
f (e
x) = (ρ(e
x))2 h(e
x)
(8.3)
where ρ and h are two C ∞ real-valued mappings such that ρ(0) = 0,
Dρ(0) , 0 and h(0) , 0.
When f is a reducible ∞-degenerate mapping, its local zero set is
that of the mapping ρ. As Dρ(0) , 0, the Implicit function theorem
shows that it consists of exactly one C ∞ curve, tangent to the characteristic Ξ at the origin (cf. (8.2)). In any system (u, v) of coordinates such
that Ξ does not coincide with the v-axis (i.e. in which Ξ is not vertical),
this curve is the graph of a C ∞ function v = v(u). In such a system of
coordinates, we can alwaus take ρ(u, v) = v − v(u): Indeed, from the
identity ρ(u, v(u)) = 0 we get
!
Z 1
(∂ρ/∂v)(u, v(u) + s(v − v(u)))ds (v − v(u))
ρ(u, v) =
0
and it suffices to replace h by the mapping
"Z
0
1
#2
(∂ρ/∂v)(u, v(u) + s(v − v(u)))ds h(u, v),
5.8. The Case of an Infinite Process.
173
(which does not vanish at the origin).
We shall see how the mapping v is related to the characteristics
Ξ, Ξ1 , · · · , Ξm , · · · With this aim, there is a choice of elements e
ξ j and
e
τ j , j ≥ 0, that is especially convenient for defining the iterates f ( j) :
Let γ0 be the slope of the characteristic Ξ in the system (u, v) (recall 222
that γ0 is well defined as a real number since Ξ is not vertical) and set
e
ξ0 = (1, γ0 ). Taking T as the v-axis, we choose e
τ0 = (0, 1). The first
iterate f (1) is entirely determined from e
ξ0 and e
τ0 and its characteristic Ξ1
is not vertical in the (u1 , v1 )-plane. Denoting by γ1 the slope of Ξ1 , we
set e
ξ1 = (1, γ1 ). Taking T 1 as the v1 -axis, we choose e
τ1 = (0, 1) which
(2)
determines the second iterate f . In general, the choice of the e
ξ j ’s and
e
τ j ’s is then as follows
e
ξ j = (1, γ j ), j ≥ 0,
(8.4)
where γ j denotes the slope of the characteristic Ξ j in the (u j , v j ) plane
and
e
τ j = (0, 1), j ≥ 0.
(8.5)
Remark 8.2. Note for every j ≥ 1 that the characteristic Ξ j depends on
e
ξℓ and e
τℓ , 0 ≤ ℓ ≤ j − 1. Thus, by modifying one of the e
ξ j ’s (or e
τ j ’s)
we affect the definition of all the characteristics of order > j. This is
to say that our further results hold with the choice (8.4)-(8.5) and with
this choice only. If any other choice of elements e
ξ j and e
τ j is made, the
relationship to the characteristics Ξ, Ξ1 , · · · , Ξm , · · · is different.
Before the first important theorem of this section, we need to establish
Lemma 8.1. Let m ≥ 1 be a given integer. We set


m−1


X
fˆ(u, v) = f u, v +
γ j u j+1  .
(8.6)
j=0
Then, the characteristics Ξ̂, Ξ̂1 , · · · , Ξ̂m−1 of fˆ, fˆ(1) , · · · , fˆ(m−1) exist
and are horizontal.
223
5. Introduction to a Desingularization Process....
174
Proof. It is clear that D fˆ(0) = 0. Now, we have
∂2 f
∂2 fˆ
∂2 f
∂2 f
(0)γ
+
(0)
=
(0)
+
2
(0)γ02 =
0
∂u∂v
∂u2
∂u2
∂v2
= D2 f (0) · (1, γ0 )2 = D2 f (0) · (e
ξ0 )2 = 0,
and
Finally, since
∂2 f
∂2 f
∂2 fˆ
(0) =
(0) + 2 (0)γ0
∂u∂v
∂u∂v
∂v
= D2 f (0) · ((1, γ0 ), (0, 1))
= D2 f (0) · (e
ξ0 ,e
τ0 ) = 0
∂2 f
∂2 fˆ
(0)
=
(0) = D2 f (0) · (0, 1)2 = D2 f (0) · (e
τ0 )2 , 0,
∂v2
∂v2
(8.7)
the above relations show that the characteristic Ξ̂ of fˆ exists and is horizontal.
Taking ê
ξ0 = (1, 0) and ê
τ0 = e
τ0 = (0, 1), the first iterate fˆ(1) is defined
by



 u12 fˆ(u1 , u1 v1 ) if u1 , 0,
ˆf (1) (u1 , v1 ) = 
1
(8.8)

2


 1 ∂ fˆ (0)v2 if u = 0.
2 ∂v2
224
By definition, one has




ˆf (1) (u1 , v1 ) = 




1
1
1
f (u1 , γ0 u1 + u1 v1 )
u21
1 ∂2 f
2
2 ∂v2 (0)v1 if u1 = 0.
if u1 , 0,
(8.9)
Thus, from (8.7)-(8.9) we deduce
ˆ(1)
f
Setting
(u1 , v1 ) = f



 m−1
X
j
γ j u1  .
u1 ,

(1) 
j=1
γ(1)
j = γ j+1 , 0 ≤ j ≤ m − 2,
(8.10)
5.8. The Case of an Infinite Process.
175
(1) the same role as γ , 0 ≤
we see that γ(1)
j
j , 0 ≤ j ≤ m − 2 play with f
j ≤ m − 1, play with f . Therefore, it follows from (8.10) that the iterate
fˆ(1) is defined from f (1) as fˆ is defined from f . The same observation
can be repeated m − 1 times and our assertion follows.
Theorem 8.1. Let f be a reducible ∞-degenerate mapping. The iterate
f (1) , · · · , f (m) , · · · being defined from e
ξ j and e
τ j , j ≥ 0 as in (8.4)-(8.5),
the function v(u) whose graph is the local zero set of f verifies
d jv
(0) = j!γ j−1 for every j ≥ 1.
du j
(8.11)
Conversely, if f is ∞-degenerate and real-analytic and the disk of
∞
P
convergence of the power series
γ j u j+1 is not {0}, the mapping f is
j=0
reducible and its local zero set coincides with the graph of the analytic
function
∞
X
v(u) =
γ j u j+1 .
(8.12)
j=0
Proof. Let m ≥ 1 be a given integer. Set
v̂(u) = v(u) −
m−1
X
γ j u j+1 .
(8.13)
j=0
With the characterization f (u, v(u)) = 0, we find
fˆ(u, v̂(u)) = 0,
225
(8.14)
where fˆ is the mapping (8.11). From Lemma 8.1, the characterictics
Ξ̂, Ξ̂1 , Ξ̂m−1 , · · · are horizontal. Thus, using Theorem 6.1,
(∂ j fˆ/∂u j )(0) = 0, 0 ≤ j ≤ 2m,
(∂ j+1 fˆ/∂u j ∂v)(0) = 0, 0 ≤ j ≤ m.
By implicit differentiation in (8.14), it easily follows that (d j v̂/du j )
(0) = 0, 0 ≤ j ≤ m. By definition of v̂ (cf. (8.13)), this is equivalent to
d jv
(0) = j!γ j−1 , 0 ≤ j ≤ m.
du j
176
5. Introduction to a Desingularization Process....
But this relation holds for any m ≥ 1, which proves (8.11).
Now, assume that f is ∞-degenerate and real-analytic and the disk
∞
P
j+1
ju is not {0}. The mapping
of convergence of the power series
j=0


∞


X

j+1
γ j u  ,
fˆ(u, v) = f u, v +
(8.15)
j=0
is real-analutic and, by the same arguments as in Lemma 8.1, we see
that fˆ is ∞-degenerate and all the characteristics Ξ̂.Ξ̂1 , · · · , Ξ̂m , · · · are
horizontal. From Theorem 6.1 again,
(∂ j fˆ/∂u j )(0) = 0 for every j ≥ 0,
(∂ j+1 fˆ/∂u j ∂v)(0) for every j ≥ 0.
As fˆ is real-analytic, we deduce that
fˆ(u, v) = v2 ĥ(u, v),
226
where ĥ is real-analytic and verifies ĥ(0) , 0 (for otherwise D2 fˆ(0)
would vanish). According to (8.15), we conclude
f (u, v) = (v −
∞
X
j=0
γ j u j+1 )2 ĥ(u, v −
∞
X
γ j u j+1 ),
j=0
a relation from which it is obvious that the local zero set of f is the graph
of the mapping v (8.12).
Remark 8.3. When f is ∞-degenerate but not real-naalytic, it may hap∞
P
pen that the disk of convergence of the power series
γ j u j+1 is not {0}
j=0
but the graph of the function v(u) =
∞
P
γj
u j+1
is not in the local zero set
j=0
of f . This is the case with
2
f (u, v) = v2 + e−1/u ,
whose characteristics Ξ, Ξ1 , · · · , Ξ1 , · · · are horizontal (i.e. γ j = 0 for
j ≥ 0) but whoce local zero set reduces to the origin. Of course, such a
mapping is not reducible.
5.8. The Case of an Infinite Process.
177
From now on, we assume that f is real-analytic and ∞-degenerate.
Let (u, v) be a system of coordinates in which the characteristic Ξ is not
vertical. The next result (Theorem 8.2) will show that the assumption
∞
P
that the disk of convergence of the series
γ j u j+1 does not reduce to
j=0
{0} in Theorem 8.1 is vacuous.
For (u, v) around the origin, we can write
X
f (u, v) =
akℓ uk vℓ ,
(8.16)
k,ℓ≥0
where a00 = a10 = a01 = 0, a02 a20 − a211 = 0. Similarly, for every m ≥ 1,
the iterate f (m) is real-analytic (which can be easily checked) and we 227
shall set
X
k ℓ
(8.17)
f (m) (um , vm ) =
a(m)
kℓ um vm ,
k,ℓ≥0
(m) 2
(m) (m)
(m)
(m)
where a(m)
00 = a10 = a01 = 0, a02 a20 − (a11 ) = 0. As Ξ is not vertical
and none of the characteristics Ξ1 , · · · , Ξm , · · · is vertical either, one has
a(m)
02 , 0. Hence, for m ≥ 0
γm = −
a(m)
11
2a(m)
02
.
Now, due to the choice of e
ξm−1 and e
τm−1 (8.4)-(8.5)
f
(m)




(um , vm ) = 


1 (m−1)
f
(um , γm−1 um + um vm ) for um , 0,
u2m
1 2 2 (m−1)
2
/∂v2m−1 )(0) = am−1
2 vm (∂ f
02 vm for um =
In particular,
a(m)
02 =
1 ∂2 f (m−1)
1 ∂2 f (m)
.
(0)
=
(0) = a(m−1)
02
2 ∂v2m
2 ∂v2m−1
Thus
a(m)
02 = a02 for every m ≥ 0,
0.
(8.18)
178
so that
228
5. Introduction to a Desingularization Process....
a(m)
γm = − 11 for every m ≥ 0.
2a02
(8.19)
(m−1)
It is easy to find a relationship betweem the coefficients a(m)
kℓ , akℓ
and γm−1 . To do this, we can formula (2.12) or identfiy the coefficients
in (8.18). We get
 k+2 
P j (m−1) j−1




(m)
ℓ ak+2− j, j γm−1 for 0 ≤ ℓ ≤ k + 2
akℓ = 
(8.20)
j=ℓ



 0 for ℓ > k + 2.
Lemma 8.2. (Weierstrass preparation theorem for functions of two variables): Let f (u, v) be an analytic function of the two complex variables
(u, v) with values in C. Assume that there is an integer ℓ ≥ 1 such that
D j f (0) = 0, 0 ≤ j ≤ ℓ − 1,
and
∂ℓ f
(0) , 0.
∂vℓ
Then, there exist analytic functions θ0 (u), · · · , θℓ−1 (u) and h(u, v)
verifying
Di θ j (0) = 0, 0 ≤ i ≤ ℓ − j − 1,
h(0) , 0,
such that
f (u, v) = (vℓ +
ℓ−1
X
θ j (u)v j )h(u, v).
j=0
Moreover, the functions θ0 , · · · , θℓ−1 and h are unique.
Proof. See e.g. Bers [3], Golubitsky and Guillemin [12].
Theorem 8.2. Every real-anaylytic mapping f such that f (0) = 0,
D f (0) = 0, D2 f (0) , 0 which is ∞-degenerate is reducible and its local
zero set is made up of exactly one analytic curve.
5.8. The Case of an Infinite Process.
229
179
Proof. From Theorem 8.1, it suffices to show that the disk of conver∞
P
γ j u j+1 does not reduce to {0}. Extending f
gence of the power series
j=0
to the complex values of u and v and by Lemma 8.2 with ℓ = 2 (recall
that (∂2 f /∂v2 )(0) , 0 from the choice of the system (u, v) of cordinates),
we find
f = ph,
(8.21)
where
p(u, v) = v2 + θ1 (u)v + θ0 (u),
is the Weierstrass polynomial of f . By the uniqueness of the decomposition (8.21) and from f = f = ph, we deduce θ1 = θ1 , θ0 = θ and h = h
so that the power series expansions of the functions θ0 , θ1 and h at the
origin have real coefficients. They are then real-analytic functions of the
variables u and v.
As p(0) = 0 and h(0) , 0, the assumption D f (0) = 0 is equivalent
to Dp(0) = 0. Hence, D2 f (0) = h(0)D2 p(0) so that D2 p(0) , 0 and
det D2 p(0) = 0 since the same relation holds with f . In addition, the
characteristic Ξ of f , and then its slope γ0 , is none other that that of
p. Denoting by p(1) the first iterate of p defined through the choice
e
ξ0 = (1, γ0 ) nad e
τ0 = (0, 1) and setting
h(1) (u1 , v1 ) = h(u1 , γ0 u1 + u1 v1 ),
one has f (1) = p(1) h(1) , with p(1) and h(1) real-analytic and h(1) (0) , 0.
By the same arguments as above, p(1) has a characteristic which is none 230
other than the characteristic Ξ1 of f (1) so that the slope γ1 can be determined through p(1) as well as through f (1) and so on. To sum up, the
Weierstrass polynomial p of f inherits the property of ∞-degeneracy
and the numbers γ j , j ≥ 0, are nothing but the slopes of the characteristics of p, p(1) , · · · so that all amounts to proving that the disk of
∞
P
convergence of the power series
γ j u j+1 does not reduce to {0} when
j=0
f (u, v) = p(u, v) =
we have
v2 + θ
1 (u)v + θ0 (u).
θ1 (u) =
∞
X
k=1
If so, and with the notation (8.16)
ak1 uk ,
(8.22)
5. Introduction to a Desingularization Process....
180
θ0 (u) =
∞
X
ak0 uk ,
(8.23)
k=2
with a02 = 1 and all the other coefficients akℓ = 0. In particular, for
ℓ ≥ 3 and an induction based on formula (8.20) shows that this remains
true at any rank m ≥ 0:
a(m)
kℓ = 0 for every k ≥ 0, every ℓ ≥ 3 and every m ≥ 0.
(8.24)
With this observation, (8.20) yields in particular
(m−1)
= ak2 for every k ≥ 0 and every m ≥ 1.
a(m)
k2 = ak2
But ak2 = 0 for k ≥ 1 and hence
= ak2 = 0 for every k ≥ 1 and very m ≥ 0.
a(m)
k2
(8.25)
Finally, with (8.20) again and (8.24), we find
(m−1)
(m−1)
γm−1 ,
a(m)
k1 = ak+1,1 + 2ak2
231
for k ≥ 0 and m ≥ 1. Owing to (8.25), this is only
(m−1)
a(m)
k1 = ak+1,1 ,
for k ≥ 1. Therefore
a(m)
k1 = ak+m,1 for every k ≥ 1 and every m ≥ 0.
(8.26)
(8.19) can be rewritten in the simple form
1
γm = − am+1,1 for every m ≥ 0.
2
As a result
∞
X
1
γ j u j+1 = − θ1 (u),
2
j=0
which proves that the disk of convergence of the power series
(8.27)
∞
P
γ j u j+1
j=0
doest not reduce to {0},
5.9. Concluding Remarks on Further Possible Developments.
181
Remark 8.4. The reader can check that proving that
limm→∞ |γm |1/(m+1) < +∞
without using the Weirstrass preparation theorem and trying to express
γm in terms of the coefficients (akℓ ) through successive applications of
formula (8.20) is inextricable.
Remark 8.5. According to (8.27) and Theorem 8.1, one must also have
2

∞

X
θ0 (u) =  γ j u j+1  ,
j=0
which a direct calculation confirms.
Remark 8.6. Even when f is not real-analytic, it may happen that the
∞
P
disk of convergence of the power series
γ j u j+1 is not {0} and the
j=0
graph of the function v(u) =
∞
P
γ j u j+1 is in the local zero set of f . For 232
j=0
instance, with
2
f (u, v) = v2 + ve−1/u ,
one has γ j = 0 for every j ≥ 0 and v = 0 is in the local zero set of f .
However, this is not always the case as already observed in Remark 8.3
5.9 Concluding Remarks on Further Possible Developments.
Formally, the method we have described can be generalized to C ∞ mappings from Rn+1 into Rn (again, the regularity of f can be weakened
without really affecting most of the results) for which there is an integer
k ≥ 2 such that
D j f (0) = 0, 0 ≤ j ≤ k − 1,
(9.1)
and the zero set of the mapping
e
ξǫRn+1 → Dk f (0) · (e
ξ)k ,
(9.2)
5. Introduction to a Desingularization Process....
182
consists of a finite number of lines through the origin. In particular,
when n = 1, this assumption is fulfilled as soon as Dk f (0) , 0 (cf.
Chapter 2 , §2). We can proceed as follows. Let e
ξ0 ǫRn+1 be a nonzero vector generating one of the lines in the zero set of the mapping
(9.2). As the analysis we made in Chapter 2 deals with each direction
separately, we deduce the existence of one and only curve in the local
zero set of f which is tangent to the line Re
ξ0 at the origin if the mapping
233
Dk f (0) · ((e
ξ)k−1 , ·)ǫL (Rn+1 , Rn )
is onto. If it is not, a first iterate (depending on e
ξ0 ) is defined by setting










(1)
f (u1 ,e
v1 ) = 








1
uk1
f (u1e
ξ0 + u1e
v1 ) if u1 , 0,
1 k
k! D f (0)
· (e
ξ0 + e
v1 )k =
1
k!
k P
k
j=1
j
Dk
(9.3)
f (0) · ((e
ξ0 )k− j (e
v1 ) j ), if u1 = 0,
where u1 ǫR and e
v1 belongs to some given n-dimensional complement
e
T of Rξ0 (in the case treated in this chapter, T is one-dimensional and
e
v1 = v1e
τ0 with v1 ǫR and e
τ0 is a fixed nonzero element of T )
(1)
The mapping f (9.3) is easily seen to be class C ∞ . If D f (1) (0)
ǫL (R × Rn , Rn ) is onto, the local zero set of f (1) is found through the
Implicit function theorem and reduces to one curve: The part of the local
zero set of f which is “along the space Re
ξ0 ” (we hope this expression
self-explanatory) is obtained by
e
x = u1e
ξ0 + u1e
v1 ,
(9.4)
with (u1 ,e
v1 ) in the local zero set of f (1) .
Remark 9.1. It is easy to check that D f (1) (0) is onto in the following
two situations
1) Dk f (0)·((e
ξ0 )k−1 , ·)ǫL (Rn+1 , Rn ) is onto: This is precisely the case
when the problem is solved through the method of Chapter 2 and
defining f (1) is unnecessary.
5.9. Concluding Remarks on Further Possible Developments.
183
2) Rank Dk f (0) · ((e
ξ0 )k−1 , ·) = n − 1 and
Dk+1 f (0) · (e
ξ0 )k+1 < RangeDk f (0) · ((e
ξ0 )k−1 , ·).
When n = 1 and k = 2, tbis is exactly the situation studied in § 3.
234
More generally, if there exists an integer k1 ≥ 1 such that
D j f (1) (0) = 0, 0 ≤ j ≤ k1 − 1,
(9.5)
(u1 ,e
v1 )ǫR × Rn → Dk1 f (1) (0) · (u1 ,e
v1 )k1 ǫRn ,
(9.6)
and the mapping
verifies the condition (R − N.D.), the local zero set of f (1) is found
through Corollary 3.1 of Chapter 2. Again, the local zero set of f “along
the space Re
ξ0 ” is obtained through the transformation (9.4).
How every, in a number of cases, it can be expected that D f (1) (0) ,
0 while D f (1) (0) is not onto either (for instance, if 1 ≤ rank Dk f (0) ·
((e
ξ0 )k−1 , ·) ≤ n − 2; cf. Remark 9.1). That is when Theorem 4.2 - that
has not been used in these lectures - has a role to play since it does not
require the first derivative to vanish at the origin.
Note that, contrary to what happens in the simplest case n = 1 and
k = 2, there is not necessarily a unique direction e
ξ0 along which Dk f (0)·
((e
ξ0 )k−1 , ·) is not onto and a first iterated f (1) must be defined for each
such direction.
Suppose now that (9.5) holds nut the mapping (9.6) does not verify
the condition (R − N.D.): If however its zero set is made up of a finite 235
number of lines through the origin, we can reduce the study of the local
zero set of f (1) to the study of the local zero set of new iterates f (2) . The
same idea can be applied - with some modifications - to the more general
case when D f (1) (0) , 0 but is not onto and Theorem 4.2 fails to hold.
Formally, the blowing-up process can be repeated under the assumption
that the zero sets of the first nonzero derivative (of order > 1 actually) at
the origin of each necessary iterate consists of a finite number of lines.
This is a serious obstacle to the development of a general theory since
it seems hard to find reasonable assumptions on f ensuring that it will a
184
5. Introduction to a Desingularization Process....
priori be so, but the method can be tried with each particular example.
Nevertheless, the above hypothesis that the zero set of the first nonzero
derivative at the origin of each necessary iterate is made up of a finite
number of lines is automatic if n = 1. It is then conceivable that this case
could be studied in a somewhat general frame work and much progress
has already been made in this direction.
Remark 9.2. More generally, for any n and when rank
Dk f (0) · ((e
ξ0 )k−1 , ·) = n − 1
(in the notation of Remark 9.1), the behaviour of the iterate f (1) and
hence of all the necessary other ones - is “as if n were 1” so that this
case also can be expected to bear some general analysis.
236
When n = 1, the general case when k is arbitrary must be handled
rather differently by using results and methods of algebraic geometry.
The relationship to singularity theory has also become clear although
some problems remain open and the related results will be found in
forthcoming publications (In the present situation when n = 1 and k = 2,
there is a simple and explicit relation between the number m of iterates
necessary for desingularizing f and the codimension of f but this is no
longer true if k ≥ 3).
Singularity theory seems to play an increasing role in the study of
perturbed bifurcation since the publication of a famous paper by Golubitsky and Schaeffer ([5]).
Appendix 1
Practical Verification of the
Conditions (R − N.D.) and
(C − N.D.) when n = 2 and
Remarks on the General
Case.
Let q = (qα )α=1,2 be a homogeneous polynomial mapping of degree k 237
on R3 with values in R2 . Given a system of coordinates (e
e1 ,e
e2 ,e
e3 ) of
3
3
e
R , we set for ξǫR
e
ξ = ξ1e
e1 + ξ2e
e2 + ξ3e
e3 ,
(A.1)
e
ξ ′ = ξ1e
e1 + ξ2e
e2 ,
(A.2)
where ξ1 , ξ2 , ξ3 ǫR and define
so that
e
ξ =e
ξ ′ + ξ3e
e3 .
(A.3)
With these notations, each polynomial qα is expressed as
qα (e
ξ) =
k
X
s=0
aα,s (e
ξ ′ )ξ3s , α = 1, 2
185
(A.4)
186
1. Practical Verification of the Conditions.....
where aα,s is a homogeneous polynomial of degree k − s. Now, note
that either the condition (R − N.D.) fails of e
e3 can be chosen so that
q1 (e
e3 ) , 0 and q2 (e
e3 ) , 0. Indeed, this is possible unless q1 q2 = 0,
namely q1 ≡ 0 or q2 ≡ 0. But, in this case, the condition (R − N.D.) is
not satisfied by q.
With this choice of e
e3 , one has
aα,k = qα (e
e3 ) , 0, α = 1, 2.
238
(A.5)
Then, q(e
ξ) = 0 if and only if the two polynomials in ξ3 of degree
exactly k
ξ3 → qα (e
ξ ′ + ξ3e
e3 ), α = 1, 2,
(A.6)
have a common real root for the prescribed value e
ξ ′ . As the leading
coefficients a1,k and a2,k are nonzero, the resultant R(e
ξ ′ ) of these two
′
e
polynomials vanishes for this values of ξ .
We examine the converse of this result. First, the resultant R can
identically vanish: It is well - known that this happens if and only if q1
and q2 have a nonconstant (homogeneous) common factor. Let us then
write
q1 = rp1 , q2 = rp2
with p1 and p2 relatively prime, deg r ≥ 1. If r vanishes at some point
e
ξǫR3 − {0}, one has
qα (e
ξ) = 0, α = 1, 2,
and
Dqα (e
ξ) = Pα (e
ξ)Dr(e
ξ), α = 1, 2.
Thus, Ker Dq(e
ξ) ⊃ KerDr(e
ξ) whose dimension is ≥ 2 and q does
not satisfy the condition (R − N.D.).
If r does not vanish in R3 − {0} (note that r does vanish in C3 − {0}
from Hilbert’s zero theorem), there is no line in the zero set of q in R3
which is in the zero set of r and we can replace q1 and q2 by p1 and p2 .
From now, on, we can then assume that q1 and q2 are relatively prime,
so that
R . 0.
239
It can be shown by simple arguments that R is a homogeneous poly-
187
nomial of degree k2 (see e.g. [16, 18]). Its zero set in Re
e1 ⊕ Re
e2 then
consists of a finite number of lines (≤ k2 ) through the origin and the
same result is true in the space Ce
e1 ⊕ Ce
e2 (of course, in this case, the
lines in the zero set of R are complex ones). For the sake of convenience,
we shall denote by K either field R of C. Given a point e
ξ ′ ǫKe
e1 ⊕ Ke
e2
′
e
such that R(ξ ) = 0, there is a finite number (≥ 1) of values ξ3 ǫC such
that q(e
ξ ′ + ξ3e
e3 ) = 0: These values are the common roots in C of the two
polynomials (A.6) (hence their number is finite). It follows that, above
each K-line in the zero set of R in Ke
e1 ⊕ Ke
e2 , there is a finite number
of K-lines in the zero set of q. However, when K = R, there lines may
not lie in R3 since, for a given e
ξ ′ ǫRe
e1 ⊕ Re
e2 in the zero set of R, some
corresponding values ξ3 such that q(e
ξ ′ + ξ3e
e3 ) = 0 may belong to C\R.
But this happens in “almost” no system (e
e1 ,e
e2 ,e
e3 ). To see this, let us
∗
consider an arbitrary change of e
e3 into e
e3 such that (e
e1 ,e
e2 ,e
e∗3 ) is a basis
of R3 . The coordinates (ξ1∗ , ξ2∗, ξ3∗ ) of any point e
ξǫR3 in this basis are
given by
 
 ∗ 
 ξ1  1 0 a ξ1 
 
 ∗  
(A.7)
 ξ1  = 0 1 b ξ2  ,
∗
0 0 c ξ3
ξ31
where (ξ1 , ξ2 , ξ3 ) are the coordinates of e
ξ in the basis (e
e1 ,e
e2 ,e
e3 ) and the
coefficients a, b and c are real with c , 0. Of course, (e
e1 ,e
e2 ,e
e3 ) and 240
(e
e1 ,e
e2 ,e
e∗3 ) are also bases in C3 and the change of coordinates is still
given by (A.7).
Let the C-lines in the zero set of q in C3 be generated by the nonzero
vectors e
ξ1 , · · · , e
ξν with ν ≤ k2 from Bezout’s theorem. Denote by
(ξ j1 , ξ j2 , ξ j3 ) and (ξ ∗j1 , ξ ∗j2 , ξ ∗j3 ) the components of e
ξ j , 1 ≤ j ≤ ν, in the
bases (e
e1 ,e
e2 ,e
e3 ) and (e
e1 ,e
e2 ,e
e∗3 ) respectively. For a fixed index 1 ≤ j ≤ ν,
the first two components ξ ∗j1 and ξ ∗j2 will have the sasme argument if and
only if (agreeing that 0 has the same argument as any complex number)
(Reξ j1 Imξ j2 − Reξ j2 Imξ j1 ) + a(Reξ j3 Imξ j2 − Reξ j2 Imξ j3 )+
+ b(Reξ j1 Imξ j3 − Reξ j3 Imξ j1 ) = 0
In other words, if ξ j1 , ξ j2 and ξ j3 do not have the same argument, the
above equality holds for a and b in a one-dimensional affine manifold
188
241
1. Practical Verification of the Conditions.....
of R2 while, from (A.7), if ξ j1 , ξ j2 and ξ j3 have the same argument,
the three components ξ ∗j1 , ξ ∗j2 and ξ ∗j3 also have the same argument. To
sum up, for a and b outside the union of a finite number ≤ ν of onedimensional affine manifolds in R2 , the following property holds: For
any 1 ≤ j ≤ ν, the two components ξ ∗j1 and ξ ∗j2 have the same argument
if and only if ξ ∗j1 , ξ ∗j2 and ξ ∗j3 have the same argument.
Now, let
e
ξ = ξ1∗e
e1 + ξ2∗e
e2 + ξ3∗e
e∗3 ,
be a nonzero element in the zero set of q in C3 with ξ1∗ , ξ2∗ ǫR. One has
e
ξ = λe
ξ j,
242
for some λǫC − {0} and some 1 ≤ j ≤ ν. Thus ξ1∗ = λξ ∗j1 , ξ2∗ = λξ ∗j2 and
ξ3∗ = λξ ∗j3 . As ξ1∗ and ξ2∗ are real by hypothesis, ξ ∗j1 and ξ ∗j2 have the same
argument (namely, -Arg λ). Hence Arg ξ ∗j3 = −Argλ too and it follows
that ξ3∗ is real.
Thus, by simply modifying e
e3 , one can assume that above each Rline in the zero set of R in Re
e1 ⊕ Re
e2 , all the corresponding R-lines in
the zero set of q in C3 lie in R3 (an equivalent way of saying that for
every e
ξ ′ ǫRe1 ⊕ Re2 such that R(e
ξ ′ ) = 0, all the solutions ξ3 ǫC of the
′
e
equation q(ξ + ξ3e
e3 ) = 0 are real). From the above proof, this property
is unchanged by an arbitrarily small change of the vector e
e3 . Of course,
above a given R-line in the zero set of R in Re
e1 ⊕ Re
e2 may lie several Rlines in the zero set of q in R3 . Again, this happens in exceptional cases
only. Indeed, the R-lines in the zero set of R are the projections along e
e3
of the lines in the zero set of q. By slightly changing e
e3 and since there
are only finitely many R-lines in the zero set of q in R3 , we can manage
so that the lines in the zero set of q and the lines in the zero set of R
are in one-to-one correspondence. We leave it to the reader to given a
rigorous proof of this result, exemplified on Figure A.1 below: Given a
basis (e
e1 ,e
e2 ,e
e3 ) of R3 and a finite number of lines in R3 which project
onto the same line of the plane Re
e1 ⊕ Re
e2 along e
e3 , any change of e
e3
into a non-collinear vector is so that the new projection onto Re
e1 ⊕ Re
e2
along e
e3 transforms these lines into the same number of distinct lines of
the plane Re
e1 ⊕ Re
e2 .
189
Remark A.1: Of course, each change of the vector e
e3 modifies the polynomial R, which is the reason why the above properties can be established.
Figure A.1:
The basis (e
e1 ,e
e2 ,e
e3 ) being now fixed so that the previous properties
hold, we are in position to give a characterization of the condition (R −
N.D.). First, as R . 0 and by a suitable choice of e
e2 without modifying
Re
e1 ⊕ Re
e2 (so that the definition of R is not affected) we may assume
R(e
e2 ) , 0. As R is homogeneous of degree k2 , e
ξ ′ = 0 is always in the 243
zero set of R. From (A.5), this value corresponds with the value e
ξ=0
in the zero set of q, a solution in which we have no interest. All then
amounts to finding the nonzero solutions of R(e
ξ ′ ) = 0. By our choice
of e2 , none of them is of the form ξ2e
e2 (i.e. ξ1 = 0) and, after dividing
2
by ξ1k , 0
ξ2
e2 ) = 0
R(e
ξ ′ ) = 0 ↔ R(e
e1 + e
ξ1
Setting τ = ξ2 |ξ1 , each real root of the polynomial
a(τ) = R(e
e1 + τe
e2 ) = 0,
(A.8)
corresponds with one and only one real line in the zero set of R in
Re
e1 ⊕ Re
e2 and hence with one and only one line in the zero set of q in
R3 . Then, the condition (R − N.D.) is equivalent to assuming that each
190
1. Practical Verification of the Conditions.....
real root of polynomial a(τ) is simple. We only sketch the proof of this
result (which uses the notion of multiplicity of intersection). For the
sake of brevity, we shall refer to the real zero set of q (resp. R) as being
the zero set of q (resp. R) in R3 (resp. Re
e1 ⊕ Re
e2 ). Similar definitions
will be used for the complex zero sets of q and R.
A real line LR in the real zero set of q (resp. R) is than the intersection of one and only one complex line LC in the complex zero set of q
(resp. R) with R3 (resp. Re
e1 ⊕ Re
e2 ). Actually, if
244
then
LR = Re
ξ0 (resp.Re
ξ0′ ),
LC = Ce
ξ0 (resp.Ce
ξ0′ ).
The line LC will be called the complex extension of LR . From our
previous results, above the extension LC of a given line LR in the real
zero set of R lines exactly one C-line in the complex zero set of q
(whose intersection with R3 is nothing but this one R-line in the real
zero set of q above LR ).
To each C-line LC in the complex zero set of q is associated a multiplicity, called the multiplicity of intersection of the surface q1 (e
ξ) = 0 and
q2 (e
ξ) = 0 along LC . The following result is true in general (cf. [11]):
The multiplicity of the root τǫC of the polynomial a(τ) (A.8) equals
the number of C-lines in the complex zero set of q which lie above the
C-line
{ξ1e
e1 + τξ1e
e2 , ξ1 ǫC},
245
in the complex zero set of R, counted with multiplicity. On the other
hand, from the condition (R − N.D.), the multiplicity of the complex
extension of any R-line in the real zero set of q happens to be one, which,
from our choice of the basis (e
e1 ,e
e2 ,e
e3 ), proves the equivalence of the
condition (R − N.D.) with the fact that each real root of the polynomial
a(τ) (A.8) is simple.
The condition (C − N.D.) also bears a similar characterization. Here,
the basis (e
e1 ,e
e2 ,e
e3 ) must be taken so that the projection onto Ce
e1 ⊕ Ce
e2
of all the C-lines in the complex zero set of q are distinct. If so, the same
191
arguments of multiplicity show that the condition (C − N.D.) is equivalent1 to assuming that each root of the polynomial a(τ) (A.8) simple (as
a result, it has exactly k2 distinct roots, each of them providing a different C-line in the complex zero set of q). It follows that the condition
(C − N.D.) amounts to assuming that the discriminant of a(τ) (A.8) is
, 0: It is a (2k2 − 1)× (2k2 − 1) determinant2 whose coefficients are completely determined by the polynomials q1 and q2 and is then particularly
simple to work out in practice.
In both cases, besides a criterion for checking the non-degeneracy
condition, we have got a practical means of calculation of (approximations of) the R-lines in the real zero set of q by calculating (approximations of) the real roots of the polynomial a(τ), an important feature for
developing the algorithmic process of Chapter 4.
REMARKS ON THE GENERAL CASE
When n is arbitrary, the problem of practical verification of the condition (R − N.D.) and calculation of (approximations of) the R-lines in
the real zero set of q is imperfectly solved. However, the practical verification of the condition (C − N.D) remains possible. Observe that it will 246
fail if and only if the (n + 1) systems of (n + 1) homogeneous equations
in (n + 1) variables



ξ) = 0ǫCn ,
 q(e
(A.9)


 △ j (ξ) = 0ǫC,
where △ j is one of the (n + 1) n × n minors of the derivative Dq(e
ξ), are
satisfied by some nonzero e
ξ ∈ Cn+1 . Each minor △ j is a polynomial
whose coefficients are polynomials in the coefficients of q. It so happens that a syatem of algebraic relations between the coefficients of q
can be found so that the system (A.9) is solvable if and only if these relations hold (cf. Hodge and Pedoe [16], where it is also shown that one
algebraic relation is sufficient). Hence the existence of a finite system of
1
As very nonconstant homogeneous polynomial vanishes at nonzero points in C3 , it
is necessary that q1 and q2 be relatively prims for the condition (C − N.D.) to hold.
2
A note by H. Gerber (Amer.Math. Monthly, 91(1984), pp.644-646) on reducing
the size of determinants for calculating resultants may be applicable here.
192
1. Practical Verification of the Conditions.....
algebraic relations between the coefficients of q which will be satisfied
Marsden and Schecter have suggested the use of Seidenberg’s algorithm
([36]) to fin such a system of algebraic relations characterizing the condition (R − N.D.) but simplifications would be desirable to get an actual
practical method of verification. Also, this approach does not yield any
means of calculation of (approximations of) the R-lines in the zero set
of q.
Appendix 2
Complements of Chapter IV
In this appendix, we give complete proofs for several results we used 247
in Chapter 4, §§5 and 6 and which were omitted there for the sake of
brevity. The notation is that of Chapter 4 and our exposition follows E1
Hajji [10].
To begin with, we make precise a few definitions and elementary
properties. Let E be a real Banach space with norm ||·|| and Q a compact
space. The Banach space C 0 (Q, E) will be equipped with the norm |·|∞,Q
defined by
ψǫC 0 (Q, E) → |ψ|∞,Q = sup ||ψ(x)||.
xǫQ
Now, if Ω denotes an open subset of some other real Banach space
and m ≥ 0 a given integer, we call C m (Ω, E) the space of mappings m
times differentiable in Ω whose derivatives of order ≤ m can be continuously extended to the closure Ω. Note that if the ambient space of Ω
is infinite-dimensional, C m (Ω, E) is not a normed space, even when Ω
is bounded. On the contrary, if Ω is bounded and its ambient space is
finite-dimensional, C m (Ω, E) is a Banach space with the norm
m
ψǫC (Ω, E) →
m
X
i
sup ||D ψ(x)|| =
i=0 xǫΩ
m
X
i=0
|Di ψ|∞,Ω .
In other words, convergence in the space C m (Ω, E) is equivalent to
convergence of the derivatives of order ≤ m in the uniform norm.
193
2. Complements of Chapter IV
194
248
Our first task is to prove, for r > 0 and δ > 0 small enough, that
the sequence (gℓ ) (resp. (Deξ gℓ )) tends to g (resp. Deξ g) in the space
C 0 ([−r, r]×△, Rn ) (resp. C 0 ([−r, r]×△, L (Rn+1 , Rn ))). Since the results
we are looking for are local, it is not restrictive to assume that O (in IV.5)
is convex and r > 0 is chosen so that te
ξǫO for every tǫ[−r, r] and every
e
ξǫ△.
Due to the definition of k, the Taylor formula shows for 0 < |t| ≤ r
and e
ξǫ△ that
Z 1
gℓ (t, e
ξ) − g(t, e
ξ) = k
(1 − s)k−1 Dk ( fℓ − f )(ste
ξ) · (e
ξ)k ds.
0
From the definitions, this remains true for t = 0 and, as ||e
ξ|| ≤ 1+δ/2
we get
δ k k
|D ( fℓ − f )|∞,O .
(A2.1)
|gℓ − g|∞,[−r,r]×△ ≤ 1 +
2
Also, for t , 0, we have
Deξ (gℓ − g)(t, e
ξ) =
k!
tk−1
D( fℓ − f )(te
ξ),
(A2.2)
while the relation gℓ (0, ·) = g(0, ·) = q for every ℓ ≥ 0 yields
Thus, if k = 1
Deξ (gℓ − g)(0, e
ξ) = 0.
δ
|Deξ (gℓ − g)|∞,[−r,r] × ∆ ≤ k(1 + )k−1 |Dk ( fℓ − f )|∞,O .
2
249
(A2.3)
(A2.4)
In order to show that the above formula remains true also when k ≥
2, it suffices to apply the Taylor formula to the term D( fℓ − f ) and to
substitute it in (A2.2); we then find, for 0 < |t| ≤ r and e
ξǫ△, that
Z 1
e
Deξ (gℓ − g)(t, ξ) = k(k − 1)
(1 − s)k−2 Dk ( fℓ − f )(ste
ξ) · (e
ξ)k−1 ds,
0
and this relation remains valid for t = 0, which establishs (A2.4) immediately. The assertion follows from (A2.1) and (A2.4) and the convergence of the sequence ( fℓ ) to f in the space C k (O, Rn ).
195
It is then easy to deduce that the sequence (Mℓ ) (respectively
(Deξ Mℓ )) tends to M (respectively Deξ M) uniformly in the set [−r, r] ×
C × △. More precisely, introducing
w(δ) = sup ||A(e
ζ0 ) − A(e
ξ0 )|| = |A − A(e
ξ0 )|∞,C ,
(A2.5)
e
ζ0 ǫC
one has
|Mℓ − M|∞,[−r,r]×C×△ ≤ (||A(e
ξ0 )|| + w(δ))|gℓ − g|∞,[−r,r]×△
(A2.6)
Besides, the same arguments leads to the inequality
|Mℓ (t, ·, ·) − M(t, ·, ·)|∞,C×△ < ||A(e
ξ0 )|| + w(δ) |gℓ (t, ·) − g(t, ·)|∞,△ ,
(A2.6)′
for every tǫ[−r, r].
Now, for (t, e
ζ0 , e
ξ)ǫ[−r, r] × C × △
||Mℓ (t, e
ζ0 , e
ξ) − M(t, e
ζ0 , e
ξ)||
||Nℓ (t, e
ζ0 , e
ξ) − N(t, e
ζ0 , e
ξ)|| ≤ 2
e
e
||M(t, ζ0 , ξ)||
Hence, from Lemma 2.1 of Chapter 4
|Nℓ − N|∞,[−r,r]×C×△ ≤
2
|Mℓ − M|∞,[−r,r]×C×△ ,
1−δ
(A2.7)
from which it follows that the sequence (Nℓ ) tends to N uniformly in the 250
set [−r, r] × C × △. The same method shows that
|Nℓ (t, ·, ·) − N(t, ·, ·)|∞,C×△ <
2
|Mℓ (t, ·, ·) − M(t, ·, ·)|∞,C×△ , (A2.7)′
1−δ
for every tǫ[−r, r].
The proof of Theorem 5.1 of Chapter 4 is based on the following
estimate.
Lemma A2.1: For every triple (t, e
ζ0 , e
ξ)ǫ[−r, r] × C × △ and every ℓǫN,
one has
||Deξ Nℓ (t, e
ζ0 , e
ξ)|| ≤
2||Deξ gℓ (0, e
ξ0 )||
3δ
w(δ)+
+
(1 − δ)
(1 − δ3 )
2. Complements of Chapter IV
196

"
#
 ||A(e
ξ0 )|| + w(δ) 
3
e
e
e

+ 2 
||g
(t,
ξ)||
+
||D
g
(t,
ξ)
−
D
g
(0,
ξ
)||
.
ℓ
0
e
e
ξ ℓ
ξ ℓ
(1 − δ)
(1 − δ)2
(A2.8)
Proof. In this proof, it will be convenient to use the following natation:
Mℓ∗ (resp. Λ∗ℓ ) will denote the value M(t, e
ζ0 , e
ξ) (resp. Deξ Mℓ (t, e
ζ0 , e
ξ)) at
0
0
some arbitrary point (t, e
ζ0 , e
ξ)ǫ[−r, r] × C × △ and Mℓ (resp. Λℓ ) the particular value Mℓ (0, e
ξ0 , e
ξ0 ) (resp. Deξ mℓ (0, e
ξ0 , e
ξ0 )). Thus, by the definition
of Mℓ
ξ0 .
(A2.9)
Mℓ0 = e
On the other hand, the calculation of Deξ M(0, e
ξ0 , e
ξ0 ) made in Chapter
4, §3 can be repeated for finding DeMℓ (0, e
ξ0 , e
ξ0 ) = Λ0 so that
ξ
ℓ
h = (e
ξ0 |e
h)e
ξ0 ,
Λ0ℓe
(A2.10)
||Mℓ0 || = 1,
(A2.11)
||Λ0ℓ || = 1.
(A2.12)
for every e
hǫRn+1 . As a result
251
With the notation introduced above, an elementary calculation gives
ζ0 , e
ξ) · e
h=
Deξ Nℓ (t, e
h
Λ∗ℓe
h|Mℓ∗ ) ∗
(Λ∗ℓe
−
Mℓ ,
||Mℓ∗ ||
||Mℓ∗ ||3
(A2.13)
for every e
hǫRn+1 . In particular, choosing (t, e
ζ0 , e
ξ) = (0, e
ξ0 , e
ξ0 ), we get
h|Mℓ0 )Mℓ0
h − (Λ0ℓe
Deξ Nℓ (0, e
ξ0 , e
ξ0 ) = Λ0ℓe
(A2.14)
Next, from (A2.9)-(A2.10), it follows that Deξ Nℓ (0, e
ξ0 , e
ξ0 ) = 0.
Therefore the relation (A2.13) is unchanged by subtracting (A2.14) from
it, which means that
  ∗

 ∗
∗)
e



 Λℓe
h|M
(Λ
h


ℓ
ℓ
0
∗
0e 0

h − 
)M
h|M
M
−
(Λ
Deξ Nℓ (t, e
ζ0 , e
ξ) · e
h =  ∗ − Λ0ℓe
ℓ
ℓ
ℓ
ℓ
||Mℓ ||
||Mℓ∗ ||3
197
for every e
hǫRn+1 . Hence the inequality
||Deξ Nℓ (t, e
ζ0 , e
ξ) · e
h|| ≤ ||
h
Λ∗ℓe
h|| + ||
− Λ0ℓe
||Mℓ∗ ||
h|Mℓ∗)
(Λ∗ℓe
||Mℓ∗||3
h|Mℓ0 )Mℓ0 ||.
Mℓ∗ − (Λ0ℓe
As a first step, we establish an estimate for the term
One has
(A2.15)
h||.
− Λ0e
Λ∗e
h
|| ||Mℓ∗ ||
ℓ
ℓ
h
Λ∗ℓe
1
− Λ0ℓe
h=
((Λ∗ )e
h + (1 − ||Mℓ∗ )Λ0ℓe
h).
∗
||Mℓ ||
||Mℓ∗ || ℓ
From (A2.11), we first find a majorisation by
1 ∗
0
∗
||Λ
−
Λ
||
+
|1
−
||M
|||
||e||.
ℓ
ℓ
ℓ
||Mℓ∗ ||
Now, since |1 − ||Mℓ∗ ||| ≤ ||Mℓ∗ − Mℓ0 || (cf. (A2.11))
||
252
h
Λ∗ℓe
1
h||.
h|| ≤
− Λ0ℓe
(||Λ∗ℓ − Λ0ℓ )|| + ||Mℓ∗ − Mℓ0 ||)||e
∗
||Mℓ ||
||Mℓ∗||
(A2.16)
Next, we look for an appropriate majorisation of the second term
appearing in the right hand side of (A2.15), namely
||
h|Mℓ∗ )
(Λ∗ℓe
||Mℓ∗ ||3
Here, we use the identity
Mℓ∗ − (Λ0ℓe
h|Mℓ0 )Mℓ0 ||.
h|Mℓ∗ ) ∗
(Λ∗ℓe
1
{((Λ∗ℓ − Λ0ℓ )e
h|Mℓ∗ )Mℓ∗+
Mℓ − (Λ0ℓe
h|Mℓ0 )Mℓ0 =
∗
||Mℓ ||
||Mℓ∗||3
h, M ∗ − M 0 )M 0 +
h|M ∗ )(M ∗ − M 0 ) + (Λ0e
+ (Λ0e
ℓ
ℓ
ℓ
ℓ
ℓ
ℓ
ℓ
ℓ
h.Mℓ0 )Mℓ0 }.
+ (1 − ||Mℓ∗ ||3 )(Λ0ℓe
As |1−||Mℓ∗ ||3 | = |1−||Mℓ∗|||(1+||Mℓ∗ ||+||Mℓ∗||2 ) and using the inequality
|1 − ||Mℓ∗||| ≤ ||Mℓ∗ − Mℓ0 ||, we obtain
||
h|Mℓ∗)
(Λ∗ℓe
||Mℓ∗ ||
Mℓ∗ − (Λ0ℓe
h|Mℓ0 )Mℓ0 || ≤
(A2.17)
2. Complements of Chapter IV
198


 1
2
1
2  ∗
∗
0
 ||Mℓ − Mℓ0 ||}||e
h||.
≤ { ∗ ||Λℓ − Λℓ || +  ∗ +
+
||Mℓ ||
||Mℓ || ||Mℓ∗ ||2 ||Mℓ∗||3
Using (A2.16) and (A2.17) in (A2.15), we arrive at




2
1
1
∗
0
 ||Mℓ∗ − Mℓ0 ||}.
||Deξ Nℓ (t, e
ζ0 , e
ξ)|| ≤
{||Λℓ − Λℓ || + 1 +
+
∗
∗
∗
||Mℓ ||
||Mℓ ||Mℓ ||2
253
In view of Lemma 2.1 of Chapter 4 and from 0 < 1 − δ < 1, this
inequality takes the simpler form
!
2
3
∗
0
∗
0
e
e
||Deξ Nℓ (t, ζ0 , ξ)|| ≤
||M − Mℓ || . (A2.18)
||Λℓ − λℓ || +
1−δ
(1 − δ)2 ℓ
At this stage, proving our assertion reduces to finding a suitable estimate for the two terms ||Λ∗ℓ − Λ0ℓ || and ||Mℓ∗ − Mℓ0 ||. We begin with a
majorisation of ||Mℓ∗ − Mℓ0 ||. By (A2.9) and the definition of the mapping
Mℓ ,
||Mℓ∗ − Mℓ0 || ≤ ||e
ξ − ξe0 || + ||A(e
ζ0 ) · gℓ (t, e
ξ)||
δ
ζ0 )||||gℓ (t, e
ξ)||.
≤ + ||A(e
2
As ||A(e
ζ0 )|| ≤ ||A(e
ξ0 )|| + w(δ) by the definition of w(δ) (of. (A2.5))
δ ξ0 )|| + w(δ) ||gℓ (t, e
ξ)||.
(A2.19)
||Mℓ∗ − Mℓ0 || ≤ + ||A(e
2
Finally, it is immediate that
ξ0 ) · e
h.
Λ∗ℓ − Λ0e
h = A(e
ζ0 )Deξ gℓ (t, e
ξ) · e
h − A(e
ξ0 )Deξ gℓ (0, e
Hence
ζ0 ) − A(e
ξ0 )||||Deξ gℓ (0, e
ξ0 )||+
||Λ∗ℓ − Λ0ℓ || < ||A(e
+ ||A(e
ζ0 )||||Deξ gℓ (t, e
ξ) − Deξ gℓ (0, e
ξ0 )||.
ξ0 )|| + w(δ) again and by the defiFrom the inequality ||A(e
ζ0 )|| < ||A(e
nition of w(δ), we find then
||λ∗ℓ − Λ0ℓ || < w(δ)||Deξ gℓ (0)||+
199
+ (||A(e
ξ0 )|| + w(δ))||Deξ gℓ (t, e
ξ) − Deξ gℓ (0, e
ξ0 )||.
(A2.20)
The combination of inequalities (A2.18) - (A2.20) yields the desired
extimate (A2.8).
Proof of Theorem 5.1: As a first step, we prove that the mapping 254
Nℓ (t, e
ξ0 , ·) is Lipschitz continuous with constant γ. The fact that N(t,
e
ζ0 , ·) maps the ball △ into itself will be shown afterwards. The starting point is the estimate (A2.8) of Lemma A2.1: From the relation
Deξ gℓ (0, e
ξ0 ) = Deξ g(0, e
ξ0 ) (= Dq(e
ξ0 ), Chapter 2, §3) and writing gℓ =
gℓ − g + g, it is easily seen that
2||Deξ g(0, e
ξ0 )||
2(||A(e
ξ0 )|| + w(δ))
3δ
+
w(δ)
+
+
||Deξ Nℓ (t, e
ζ0 ,e
ǫ )|| ≤
3
1−δ
1−δ
(1 − δ)
"
#
3
||g(t, e
ξ)|| + ||Deξ g(t, e
ξ) − Deξ g(0, e
ξ0 )|| +
(1 − δ)2
2(||A(e
ξ0 )|| + w(δ))
+
1−δ
#
"
3
||(gℓ − g)(t, e
ξ)|| + ||Deξ (gℓ − g)(t, e
ξ)|| .
(1 − δ)2
Given any constant γ > 0, the sum of the first three terms of the
above inequality can clearly be made ≤ γ/2 provided r > 0 and δ with
0 < δ < 1 are taken small enough. Fixing then r, and δ (for the time
being), the last term is uniformly bounded by
#
"
2(||A(e
ξ0 )|| + w(δ))
3
|gℓ − g|∞,[−r,r]×△ + |Deξ (gℓ − g)|∞,[−r,r]×△ .
1−δ
(1 − δ)2
For ℓ large enough, say ℓ ≥ ℓ0 , it can then be made ≤ γ/2 as well.
By the mean value theorem, the mapping N(t, e
ζ0 , ·) is then Lipschitz
continuous with constant γ in the ball △ for every pair (t, e
ζ0 )ǫ[−r, r] × C
and every ℓ ≥ ℓ0 . This property is not affected by shrinking r > 0 and
0 < δ < 1 arbitrarily. Applying then Theorem 3.1 to each mapping
Nℓ , 0 ≤ ℓ ≤ ℓ0 − 1 (thus a finite number of times), it is not restrictive
to assume that r > 0 and δ with 0 < δ < 1 are such that the mapping 255
200
2. Complements of Chapter IV
Nℓ (t, e
ζ0 , ·) is a contraction with constant γ in the ball △ for every ℓǫN.
Again, this property remains true if r > 0 is arbitrarily diminished. Let
us then fix δ as above. We shall show that Nℓ (t, e
ζ0 , ·) maps the ball △
into itself for every ℓǫN after modifying r > 0 if necessary.
For every triple (t, e
ζ0 , e
ξ)ǫ[−r, r] × C × △ and since N(0, e
ζ0 , e
ξ0 ) = e
ξ0
(cf. relation (2.10) of Chapter 4), one has
||Nℓ (t, e
ζ0 , e
ξ) − e
ξ0 || ≤ ||Nℓ (t, e
ζ0 , e
ξ) − Nℓ (t, e
ζ0 , e
ξ0 )||+
+ ||Nℓ (t, e
ζ0 , e
ξ0 ) − N(t, e
ζ0 , e
ξ0 )|| + ||N(t, e
ζ0 , e
ξ0 ) − N(0, e
ζ0 , e
ξ0 )||.
The first term in the right hand side is bounded by γ||e
ξ −e
ξ0 || ≤ γδ/2.
e
As γ < 1 and further the mapping N(·, ·, ξ0 ) is uniformly continuous on
the compact set [−r, r] × C, we may assume that r > 0 is small enough
fro the inequality
δ
||N(t, e
ζ0 , e
ξ0 ) − N(0, e
ζ0 , e
ξ0 )|| ≤ (1 − γ) ,
4
to hold. The term ||Nℓ (t, e
ζ0 , e
ξ0 ) − N(t, e
ζ0 , e
ξ0 )|| is bounded by
|Nℓ − N|[−r,r]×C×△
256
and can therefore be made ≤ (1 − γ)δ/4 for ℓ large enough, say ℓ ≥ ℓ0 .
Thus
δ
||Nℓ (t, e
ζ0 , e
ξ) − e
ξ0 || ≤ ,
2
for every triple (t, e
ζ0 , e
ξ)ǫ[−r, r] × C × △ and every ℓ ≥ ℓ0 . This property
notbeing affected by diminishing r > 0 arbitrarily (without (modifying
δ, of course) and observing that Nℓ (0, e
ζ0 , e
ξ0 ) = e
ξ0 for every e
ζ0 ǫC, let us
write for 0 ≤ ℓ ≤ ℓ0 − 1
||Nℓ (t, e
ζ0 , e
ξ) − e
ξ0 || ≤ ||Nℓ (t, e
ζ0 , e
ξ) − Nℓ (t, e
ζ0 , e
ξ0 )||+
+ ||Nℓ (t, e
ζ0 , e
ξ0 ) − Nℓ (0, e
ζ0 , e
ξ0 )||.
The first term in the right hand side is bounded by γ||e
ξ −e
ξ0 || ≤
γδ/2. Owing to the uniform continuity if the mapping Nℓ (·, ·, e
ξ0 ) on the
201
compact set [−r, r] × C, we may assume that r > 0 is small enough for
the inequality
||Nℓ (t, e
ζ0 , e
ξ0 ) − Nℓ (0, e
ζ0 , e
ξ0 )|| ≤ (1 − γ)δ/2,
to hold for 0 ≤ ℓ ≤ ℓ0 − 1 and, for these indices as well, we get
δ
||Nℓ (t, e
ζ0 , e
ξ) − e
ξ0 || ≤ ,
2
which completes the proof.
In chapter 4, Theorem 5.1 has been used for proving the convergence
of the sequence e
xℓ = te
ζℓ (where e
ζℓ+1 = Nℓ (t, e
ζ0 , e
ζℓ )) to e
x = te
ξ (where e
ξ is
e
the unique fixed point of the mapping N(t, ζ0 , ·) in the ball △). We shall
now give an estimate of the rate of convergence of the sequence (e
xℓ ) to
e
x under suitable assumptions on the rate of convergence of the sequence
( fℓ ) to f in appropriate spaces.
Theorem A2.1 : (i) Assume that the sequence ( fℓ ) tends to f geometrically in the space C k (O, Rn ). Then, there are constant 0 < γ′ < 1 and
K > 0 such that, for every tǫ[−r, r]
||e
xℓ − e
x|| ≤ K|t|γ′ℓ ,
(A2.21)
||e
xℓ − e
x ≤ Kγ′ℓ .
(A2.22)
for every ℓ ≥ 0.
(ii) Assume only that the sequence ( fℓ ) tends to f in the space C k (O, 257
Rn ), the convergence being geometrical in the space C k−1 (O, Rn ). Then,
there are constant 0 < γ′ < 1 and K > 0 such that, for every tǫ[−r, r]
Proof. From the relation
e
ζℓ+1 − e
ξ = Nℓ (t, e
ζ0 , e
ζℓ ) − N(t, e
ζ0 , e
ξ)
= Nℓ (t, e
ζ0 , e
ζℓ ) − Nℓ (t, e
ζ0 , e
ξ) + Nℓ (t, e
ζ0 , e
ξ) − N(t, e
ζ0 , e
ξ),
and from Theorem 5.1 of Chapter 4, we get
||e
ζℓ+1 − e
ξ|| < γ||e
ζℓ − e
ξ|| + ||Nℓ (t, e
ζ0 , e
ξ) − N(t, e
ζ0 , e
ξ)||.
2. Complements of Chapter IV
202
Hence
||e
ζℓ+1 − e
ξ|| ≤ γ||e
ζℓ − e
ξ|| + |Nℓ (t, ·, ·) − N(t, ·, ·)|∞,C×△ ,
(A2.23)
for every tǫ[−r, r]. To prove (i), observe that |Nℓ (t, ·, ·) − N(t, ·, ·)|∞,C×△ ≤
|N − Nℓ |∞,[−r,r]×C×△ . From (A2.6) - (A2.7) and (A2.1)
2(||A(e
ξ0 )|| + w(δ))
|gℓ − g|∞,[−r,r]×△
||e
ζℓ+1 − e
ξ|| ≤ γ||e
ζℓ − e
ξ|| +
1−δ
2(||A(e
ξ0 )|| + w(δ)) δ k k
≤ γ||e
ζℓ − e
ξ|| +
1+
|D ( fℓ − f )|∞,O .
1−δ
2
As the sequence ( fℓ ) tends to f geometrically in the space C k (O, Rn )
by hypothesis and after increasing 0 < γ < 1 if necessary, there is a
constant C > 0 such that
258
||e
ζℓ+1 − e
ξ|| ≤ γ||e
ζℓ − e
ξ|| + Cγℓ .
By a simple induction argument, we find
Thus
||e
ζℓ+1 − e
ξ|| ≤ γℓ+1 ||e
ζ0 − e
ξ|| + (ℓ + 1)Cγℓ .
!
ℓC
,
||e
ζℓ − e
ξ|| ≤ γℓ ||e
ζ0 − e
ξ|| +
γ
for every ℓ ≥ 0. As e
ζ0 ǫC ⊂ △ and e
ξǫC ⊂ △ and the ball △ has diameter
δ<1
!
ℓC
ℓ
e
e
||ζℓ − ξ|| ≤ γ 1 +
.
(A2.24)
γ
Multiplying by |t|, we see that
!
ℓC
||e
xℓ − e
x|| ≤ |t|γ 1 +
.
γ
ℓ
Choosing γ < γ′ < 1, inequality (A2.21) follows with
γ
K = sup ′
ℓ≥0 γ
!ℓ
!
ℓC
1+
< +∞.
γ
203
To prove (ii), we use the same method, replacing the relations (A2.6)
and (A2.7) by (A2.6)′ and (A2.7)′ . With (A2.23), we get
2(||A(e
ξ0 )|| + w(δ))
|gℓ (t, ·) − g(t, ·)|∞,△ .
||e
ζℓ+1 − e
ξ|| ≤ γ||e
ζℓ − e
ξ|| +
1−δ
Now, note that
|gℓ (t, ·) − g(t, ·)|∞,△ ≤
k δ k−1 k−1
1+
|D ( fℓ − f )|∞,O .
|t|
2
Indeed, this is obvious if k = 1. If k ≥ 2, the inequality follows from
the relation
Z
k(k − 1) 1
e
e
(1 − s)k−1 Dk−1 ( fℓ − f )(ste
ζ) · (e
ζ)k−1 ds.
gℓ (t, ζ) − g(t, ζ) =
t
0
Arguing as before, we find the analogue of (A2.24), namely
!
ℓC
ℓ
e
e
||ζℓ − ξ|| ≤ γ 1 +
,
γ|t|
259
for every ℓ ≥ 0. Multiplying by |t| > 0, we obtain
!
!
ℓC
ℓC
ℓ
ℓ
||e
xℓ − e
x|| ≤ γ |t| +
≤γ r+
.
γ
γ
Choosing γ < γ′ < 1, the inequality (A2.22) follows with
γ
K = sup ′
ℓ≥0 γ
and remains valide for t = 0.
!ℓ
!
ℓC
r+
< +∞,
γ
We shall now prove the results about convergence in the spaces of
type C k that we used in §6 of Chapter 4. Recall that given a real Banach
e k ≥ 1, where
e and a mapping Φ(= Φ(e
space Z
x,e
z))ǫC k (O × B(0, ρ), Z),
B(0, ρ) denotes the closed ball with radius ρ > 0 centered at the origin of
e verifying Φ(0) = 0 and Dez Φ(0) = 0, it is possible to shrink ρ > 0 and
Z,
2. Complements of Chapter IV
204
the neighbourhood O so that, given any arbitrary constant 0 < β < 1,
one has
||Dez Φ(e
x,e
z)|| ≤ β
(A2.25)
x, ·) is a contraction with
for every (e
x,e
z)ǫO × B(0, ρ) and the mapping Φ(e
constant β from B(0, ρ) to itself for every e
xǫO. If so, for e
xǫO, the sequence



ϕ0 (e
x) = 0,
e


e
ϕℓ+1 (e
x) = Φ(e
x, e
ϕℓ (e
x)), ℓ ≥ 0,
260
e with values
is well defined, each mapping e
ϕℓ being in the space C k (O, Z)
in B(0, ρ). We already know that this sequence converges pointwise to
e characterized by
the mapping e
ϕǫC k (O, Z)
e
ϕ(e
x) = Φ(e
x, e
ϕ(e
x)),
for every e
xǫO. In §6 of Chapter 4, we used the fact that the sequence
e the convergence being geometrical
(e
ϕℓ ) tends to e
ϕ in the space C k (O, Z),
k−1
e
in the space C (O, Z). Proving this assertion will take us a certain
amount of time. To begin with, we show that
e
ϕ in the space C 0 (O, Z).
Lemma A2.2: The sequence (e
ϕℓ ) tends to e
Proof. Let ℓ ≥ 1 be fixed. For every e
xǫO
e
ϕℓ+1 (e
x) − e
ϕℓ (e
x) = Φ(e
x, e
ϕℓ (e
x)) − Φ(e
x, e
ϕℓ−1 (e
x))
Z 1
=
x) + s(e
ϕℓ (e
x) − e
ϕℓ−1 (e
x)))·
Dez Φ(e
x, e
ϕℓ−1 (e
0
Due to (A2.25) we obtain
(e
ϕℓ (e
x) − e
ϕℓ−1 (e
x))ds.
||e
ϕℓ+1 (e
x) − e
ϕℓ (e
x)|| ≤ β||e
ϕℓ (e
x) − e
ϕℓ−1 (e
x)||,
for every e
xǫO. Thus
ϕℓ − e
ϕℓ−1 |∞,O .
|e
ϕℓ+1 − e
ϕℓ |∞,O ≤ β|e
205
Hence, the sequence (e
ϕℓ ) tends (geometrically) to a limit in the space
e Of course, this limit must be the pointwise limit e
C 0 (O, Z).
ϕ and the
proof is complete.
e the space of j-linear mapFor j ≥ 0, let us denote by L j (Rn+1 , Z)
n+1
e with the usual abuse of notation L0 (Rn+1 , Z)
e =
pings from R
into Z
e
Z. Also, recall the canonical isomorphism
e ≃ L (Rn+1 , Li (Rn+1 , Z)),
e
Li+1 (Rn+1 , Z)
(A2.26)
which will be repeatedly used in sequel.
261
e
We shall denote by λi the generic element of the space Li (Rn+1 , Z).
n+1
e = Z,
e this means, in particular,
Due to the identification L0 (R , Z)
e
that we shall identify the element e
zǫ Z with λ0 so that the assumption
Dex Φ(0) = 0 will be rewritten as
Dλ0 Φ(0) = 0.
(A2.27)
Now, setting Φ = Φ0 , introduce the mappings
Φ j : O × B(0, ρ) ×
j
Y
i=1
e → L j (Rn+1 , Z),
e 1 ≤ j ≤ k,
Li (Rn+1 , Z)
by
∂Φ j−1
(e
x, λ0 , · · · , λ j−1 ) +
(A2.28)
∂x
j−1
X
∂Φ j−1
(e
x, λ0 , · · · , λ j−1 )λi+1 ,
+
∂λi
i=0
Φ j (e
x, λ0 , · · · , λ j ) =
where the term
mappings
and
∂Φ j−1
x, λ0 , · · ·
∂λi (e
, λ j−1 )λi+1 is the product of the linear
∂Φ j−1
e L j−1 (Rn+1 , Z))
e
(e
x, λ0 , · · · , λ j−1 )ǫL (Li (Rn+1 , Z),
∂λi
e
λi+1 ǫL (Rn , Li (Rn+1 , Z)).
2. Complements of Chapter IV
206
Remark A2.1: Observe, in particular, that the only term involving λ j in
∂Φ j−1 e
(Z, λ0 , · · · , λ j−1 )λ j it follows
the definition of Φ j , j ≥ 1, is the term ∂λ j−1
that Φ j (e
x, λ0 , · · · , λ j ) is linear with respect to λ j when j ≥ 1.
The importance of the mappings Φ j , 0 ≤ j ≤ k lies in the fact that
262
D je
ϕ(e
x) = Φ j (e
x, e
ϕ(e
x), De
ϕ(e
x), · · · , D je
ϕ(e
x)), 0 ≤ j ≤ k
(A2.29)
D je
ϕℓ+1 (e
x) = Φ j (e
x, e
ϕℓ (e
x), De
ϕℓ (e
x), · · · , D je
ϕℓ (e
x)), 0 ≤ j ≤ k,
(A2.30)
for every e
xǫO and, for every ℓ ≥ 0
for every xǫO. These properties can be immediately checked by induction. Also, it is clear that the mapping Φ j is of class C k− j and, for
1 ≤ j ≤ k, Φ j is of class C ∞ with respect to (λ1 , · · · , λ j ). We now
establish two simple preliminary lemmas.
Lemma A2.3: Given any index 0 ≤ j ≤ k, one has
∂Φ j
(0, 0, λ1 , · · · , λ j ) = 0,
∂λ j
for every (λ1 , · · · , λ j )ǫ
j
Q
i=1
(A2.31)
e
Li (Rn+1 , Z).
Proof. For j = 0, the result is nothing but (A2.27). For j ≥ 1, it follows
e
from Remark A2.1 that for every µ j ǫL (Rn+1 , Z)
∂Φ j−1
∂Φ j
(e
x, λ0 , · · · , λ j ) · µ j =
(e
x, λ0 , · · · , λ j−1 )µ j .
∂λ j
∂λ j−1
Choosing e
x = 0, λ0 = 0, the above relation yields the desired result
by an indeuction argument.
Lemma A2.4 : Let the index 0 ≤ j ≤ k be fixed. If the sequence (e
ϕℓ )
j
e
tends to e
ϕ in the space C (O, Z) (which is already known for j = 0), the
set
[
e
Λj =
D je
ϕℓ (O) ⊂ L j (Rn+1 , Z),
(A2.32)
ℓ≥0
is compact and one has
D je
ϕ(O) ⊂ Λ j .
(A2.33)
207
263
Proof. A sequence in the set
S
(p)
ℓ≥0
D je
ϕℓ (O) is of the form (λ j ) p>0 where
for each pǫN there is an index ℓ = ℓ(p) and an element e
x p ǫO such
(p)
j
that λ j = D e
ϕℓ(p) (e
x p ). If, for infinitely many indices p, namely for a
subsequence (pm ), the index ℓ(pm ) equals some fixed value ℓ, one has
(pm )
λj
= D je
ϕℓ (e
x pm ).
As the set O is compact and after extracting a subsequence, we may
assume that there is e
xǫO such that the sequence (e
x pm ) tends to e
x. By
(pm )
j
continuity of the mapping D e
ϕℓ , we deduce that the sequence λ j tends
to D je
ϕℓ (e
x) as m tends to +∞.
Now, assume that the mapping p → ℓ(p) takes only finitely many
times any given value ℓ. Then, there is a subsequence ℓ(pm ) which is
strictly increasing and thus tends to +∞ as m tends to +∞. For every m,
let us write
D je
ϕℓ(pm ) (e
x pm ) = D je
ϕℓ(pm ) (e
x pm ) − D je
ϕ(e
x pm ) + D je
ϕ(e
x pm ).
Again, in view of the compactness of the set O, we may assume
ϕℓ )ℓ≥0 tends
that the sequence (e
x pm ) tends to e
xǫO. As the sequence (D je
j
0
n+1
e
to D e
ϕ in the space C (O, L j (R , Z)), by hypothesis, the same property holds for the subsequence (D je
ϕℓ(pm ) )m≥0 . Therefore, in the space
e
L j (Rn+1 , Z)
h
i
ϕℓ(pm ) (e
x pm ) − D je
ϕ(e
x pm ) = 0
lim D je
m→+∞
On the other hand, by the continuity of the mapping D je
ϕ, we get
lim D je
ϕ(e
x pm ) = D je
ϕ(e
x).
m→+∞
(pm )
As a result, the sequence (λ j
)m≥0 tends to D je
ϕ(e
x).
To sum up, we have shown that every sequence (λ(p)
j ) p≥0 of the set
S j
n+1
e and hence is
De
ϕℓ (O) has a cluster point in the space L j (R , Z)
ℓ≥0
relatively compact, which proves (A2.32). The relation (A2.33) is now
obvious since D je
ϕ(e
x) is the limit of the sequence (D je
ϕℓ (e
x))ℓ≥0 for every
e
xǫO.
264
2. Complements of Chapter IV
208
In oure new notation, the condition (A2.25) can be rewritten as
||
∂Φ0
(e
x, λ0 )|| ≤ β,
∂λ0
(A2.34)
for every (e
x, λ0 )ǫO × B(0, ρ). It will be essential in the sequel to have
an appropriate generalization of it, which we establish in the following
lemma.
Lemma A2.5 : Let the index 0 ≤ j ≤ k be fixed. If the sequence (e
ϕℓ )
e (which is already known for j = 0)
tends to e
ϕ in the space C j (O, Z)
there is no loss of generality in assuming that
||
∂Φ j
(e
x, λ0 , · · · , λ j )|| ≤ β,
∂λ j
(A2.35)
for every (e
x, λ0 , · · · , λ j )ǫO × Λ0 × · · · × Λ j .1
265
Proof. First from the previous lemma, the set O×Λ0 ×· · ·×Λ j is compact
and the restriction of the continuous mapping (∂Φ j /∂λ j ) to O × Λ0 ×
· · · × Λ j is then uniformly continuous. Next, since the mappings e
ϕ and
e
ϕℓ , ℓ ≥ 0, take their values in the ball B(0, ρ), one has Λ0 ⊂ B(0, ρ). Due
to Lemma A2.3, and the uniform continuity of the mapping (∂Φ j /∂λ j ),
there is a neighbourhood O ′ ⊂ O of the origin inh Rn+1 and 0 <i ρ′ ≤ ρ
′
such that (A2.35) holds for (e
x, λ0 , · · · , λ j )ǫO × Λ0 ∩ B(0, ρ′ ) × Λ1 ×
· · · × Λ j . Let then ρ′ be fixed. Arguing as in §6 of Chapter 4, we can
′
ϕ, ℓ ≥ 0, take their
shrink the neighbourhood O so that the mappings e
′
values in the ball B(0, ρ′ ) for xǫO . Of course, the sequence (e
ϕℓ ) still
′
S
e and the sets Λ′ =
tends to e
ϕ in the space C j (O, Z)
Di e
ϕℓ (O ), 0 ≤ i ≤
i
ℓ≥0
Λ′i
j, are compact (Lemma A2.4) with
⊂ Λi . Also, Λ′0 ⊂ B(0, ρ′ ) so that
′
inequality (A2.35) is a fortiori valid with (e
x, λ0 , · · · , λk )ǫO × Λ′0 × · · · ×
Λ′j . In other words, none of the properties we have proved is affected
if we restrict ourselves to the neighbourhood O ′ instead of O and, in
addition, inequality (A2.35) holds.
1
For j = 0, (A2.35) is nothing but (A2.34) for the elements (e
x, λ0 )ǫO × Λ0 , thus a
weak form of (A2.34).
209
For 0 ≤ j ≤ k − 1 and e
xǫO, we shall set
∂Φ j
(e
x, e
ϕ(e
x), De
ϕ(e
x), · · · , D je
ϕ(e
x)),
∂e
x
∂Φ j
(e
x, e
ϕℓ (e
x), De
ϕℓ (e
x), · · · , D je
ϕℓ (e
x)) for ℓ ≥ 0.
biℓ (e
x) =
∂e
x
b j (e
x) =
Similarly, for 0 ≤ j ≤ k − 1, 0 ≤ i ≤ j and e
xǫO, set
∂Φ j
(e
x, e
ϕ(e
x), De
ϕ(e
x), · · · , D je
ϕ(e
x)),
∂λi
∂Φ j
j
Biℓ (e
x) =
(e
x, e
ϕℓ (e
x), De
ϕℓ (e
x), · · · , D je
ϕℓ (e
x)) for ℓ ≥ 0.
∂λi
j
Bi (e
x) =
(A2.36)
(A2.37)
(A2.38)
(A2.39)
j
x) (resp. bℓ (e
x)) is an element of the space L (Rn+1 , 266
For every e
xǫO, b j (e
j
e and B j (e
L j (Rn+1 , Z))
x)) is an element of the space L (Li
i x) (resp. Biℓ (e
n+1
n+1
e
e
(R , Z), L j (R , Z)).
Lemma A2.6: (i) Assume that k ≥ 2 and 0 ≤ j ≤ k − 2. Then, if the see the sequence
quence (e
ϕℓ ) tends to e
ϕ geometrically in the space C j (O, Z),
j
j
(bℓ )ℓ≥0
tends
to
b
geometrically
in
the
space
j
0
n+1
n+1
e and the sequence (B )ℓ≥0 tends to B j geC (O, L (R , L j (R , Z)))
i
iℓ
e L j (Rn+1 , Z))).
e
ometrically in the space C 0 (O, L (Li (Rn+1 , Z),
(ii) Assume only k ≥ 1 and 0 ≤ j ≤ k − 1 and that the sequence (e
ϕℓ )
e Then, the sequence (b j )ℓ≥0 tends to
tends to e
ϕ in the space C j (O, Z).
ℓ
e and the sequence (B j )ℓ>0
b j in the space C 0 (O, L (Rn+1 , L j (Rn+1 , Z)))
iℓ
j
e L j (Rn+1 , Z))).
e
tends to Bi in the space C 0 (O, L (Li (Rn+1 , Z),
Proof. First, as the mapping Φ j is of class C k− j , the mappings b j , bℓj , Biℓj
j
and Bi are of class C 1 in the case (i) and of class C 0 in the case (ii).
Throughout this proof and for the sake of conveience, we shall set for
e
xǫO
V j (e
x) = (e
ϕ(e
x), De
ϕ(e
x), · · · , D je
ϕ(e
x))
(A2.40)
and
j
Vℓ (e
x) = (e
ϕℓ (e
x), De
ϕℓ (e
x), · · · , D je
ϕℓ (e
x)).
(A2.41)
2. Complements of Chapter IV
210
This allows to write the relations (A2.36) - (A2.39) in the form
∂ϕ j
(e
x, V j (e
x)),
∂e
x
∂Φ j
(e
x, Vℓj (e
x)),
bℓj (e
x) =
∂e
x
∂Φ j
j
Bi (e
x) =
(e
x, V j (e
x)),
∂λi
∂Φ j
j
Biℓ (e
x) =
(e
x, V − ℓ j (e
x)).
∂λi
b j (e
x) =
(A2.42)
(A2.43)
(A2.44)
(A2.45)
267
Before proving (i), note that, for e
xǫO and 0 ≤ s ≤ 1, the combinations
j
V j (e
x) + s(Vℓ (e
x) − V j (e
x)),
belong to the product Λc0 × · · · × Λcj of the closed convex hulls 2 Λci of
the compact sets Λi (cf. Lemma A2.4). As the closed convex hull of a
compact set is by a classical result from topology, the product Λc0 × · · · ×
ϕ and e
ϕℓ for
Λcj is compact. On the other hand, sinced the mappings e
ℓ ≥ 0 take their values in the ball B(0, ρ), we deduce that Λ0 and hence
Λc0 is contained in B(0, ρ). Therefore, for every 0 ≤ i ≤ j ≤ k − 2, an
expression such as
∂2 Φ j
j
(e
x, V j (e
x) + s(Vℓ (e
x) − V j (e
x))),
∂e
x∂λi
xǫO and due to the Taylor
is well defined for e
xǫO and 0 ≤ s ≤ 1. For e
formula (which can be used in view of the above observation) we get
bℓj (e
x)−b j (e
x) =
j Z
X
i=0
0
1
∂2 Φ j
(e
x, V j (e
x)+s(Vℓj (e
x)−V j (e
x)))·(Die
ϕℓ (e
x)−Di e
ϕ(e
x))ds.
∂e
x∂λ j
From the compacteness of the set O × Λc0 × · · · × Λcj , the quantities
268
∂2 Φ
x, λ0 , · · · , λ j )|| are bounded by a constant K > 0 for (e
x, λ0 , . . . ,
|| ∂ex∂λji (e
2
hull.
Recall that the closed convex hull of a set is defined as the closure of its convex
211
λ j )ǫO × Λc0 × · · · × Λcj and 0 ≤ i ≤ j. Thus
j
||bℓ (e
x) − b j (e
x)|| ≤ K
j
X
i=0
||Di e
ϕℓ (e
x) − Die
ϕ(e
x)||.
As the sequence (Di ϕℓ )ℓ≥0 tends to Die
ϕ geometrically in the space
0
n+1
e
C (O, Li (R , Z)) for every 0 ≤ i ≤ j by hypothesis, there exist q with
0 < q < 1 and a constant C > 0 such that
||Die
ϕℓ (e
x ) − Di e
ϕ(e
x)|| ≤ Cqℓ ,
for every 0 ≤ i ≤ j and every e
xǫO. Hence
j
||bℓ (e
x) − b j (e
x)|| ≤ ( j + 1)KCqℓ ,
for every e
xǫO and then
j
sup ||bℓ (e
x) − b j (e
x)|| ≤ ( j + 1)KCqℓ .
e
xǫO
j
The above inequality proves that the sequence (bℓ )ℓ≥0 tends to b j
B)). Similar argugeometrically in the space C 0 (O, L (Rn+1 , Li (Rn+1 , e
j
j
ments can be used for showing that the sequence (Biℓ )ℓ≥0 tends to Bi gee L j (Rn+1 , Z))),
e 0 ≤ i ≤ j.
ometrically in the space C 0 (O, L (Li (Rn+1 , Z),
Now, we prove part (ii) of the statement. Here, the Taylor formula
is not available because of the lack of regularity for j = k − 1. Actually, as we no longer interested in the rate of convergence, the resulta can be easily deducaed from the uniform continuity of the continuous mapping (∂Φ j /∂e
x) and (∂Φ j /∂λi ), 0 ≤ i ≤ j on the compact set
O × Λ0 × · · · × Λ j together with the uniform convergence (i.e. in the 269
e of the sequence (Die
space C 0 (O, Li (Rn+1 , Z))
ϕℓ )ℓ≥0 to the mapping Die
ϕ
for 0 ≤ i ≤ j from which the uniform convergence of the sequence
j
(Vℓ )ℓ≥0 to the mapping V j is derived.)
§
We are now in a position to prove an important part of the
results announced before.
2. Complements of Chapter IV
212
Theorem A2.2 : After shrinking the meighbourhood O is necessary,
the sequence (e
ϕℓ ) tends to the mapping e
ϕ geometrically in the space
e
C k−1 (O, Z).
Proof. Equivalently, we have to prove for 0 ≤ j ≤ k − 1 the sequence
e
(D je
ϕℓ )ℓ≥0 tends to D je
ϕ geometrically in the space C 0 (O, L j (Rn+1 , Z)).
This result has already been proved for j = 0 in Lemma A2.2 and we
can thus assume j ≥ 1 and k ≥ 2. We proceed by induction and therefore
assume that the sequemce (Die
ϕℓ )ℓ≥0 tends to Die
ϕ geometrically in the
0
n+1
e
space C (O, Li (R , Z)) for 0 ≤ i ≤ j − 1.
Let ℓ ≥ 0 be fixed. From (A2.29) and (A2.30), one has for e
xǫO
D je
ϕℓ+1 (e
x) − D je
ϕ(e
x) = Φ j (e
x, e
ϕℓ (e
x), De
ϕℓ (e
x), . . . , D je
ϕℓ (e
x))
− Φ j (e
x, e
ϕ(e
x), De
ϕ(e
x), . . . , D je
ϕ(e
x)).
By definition of the mapping Φ j (cf. (A2.48)) and in the notations
(A2.42) - (A2.45), this identity can be rewritten as
j−1
D je
ϕℓ+1 (e
x) − D j e
ϕ(e
x) = bℓ (e
x) − b j−1 (e
x)
+
j−1 h
X
i=0
270
i
j−1
j−1
Biℓ (e
x)Di+1 e
ϕℓ (e
x) − Bi (e
x)Di+1 e
ϕ(e
x) .
Recall in this formula that the element Di+1 e
ϕℓ (e
x) (resp. Di+1 e
ϕ(e
x))
n+1
n+1
e and the operation
is considered as an element of L (R , Li (R , Z))
Biℓj−1 (e
x)Di+1 e
ϕℓ (e
x) (resp. Bij−1 (e
x)Di+1 e
ϕ(e
x)) denotes composition of linear
mappings. Thus, we can as well write
j−1
x) − b j−1 (e
x)
D je
ϕℓ+1 (e
x) − D j e
ϕ(e
x) = bℓ (e
j−1
X
j−1
j−1
+
x) − Bi (e
x))Di+1 e
ϕ(e
x)
(Biℓ (e
i=0
+
j−1
X
i=0
We deduce the inequality
j−1
j−1
Biℓ (e
x)(Di+1 e
ϕℓ (e
x) − Di+1 e
ϕ(e
x)).
||D je
ϕℓ+1 (e
x) − D j e
ϕ(e
x)|| ≤ ||bℓ (e
x) − b j−1 (e
x)||
213
+
+
j−1
X
i=0
j−1
X
i=0
j−1
j−1
||Biℓ (e
x) − Bi (e
x)||||Di+1 e
ϕ(e
x)|| (A2.46)
j−1
||Biℓ (e
x)||||Di+1 e
ϕℓ (e
x) − Di+1 e
ϕ(e
x)||.
It is essential to pay special attention to the term corresponding with
i = j − 1 in the last sum, namely ||B j−1
(e
x)||||D j e
ϕℓ (e
x) − D je
ϕ(e
x)||. From
j−1,ℓ
(A2.39)
j−1
B j−1,ℓ (e
x) =
∂Φ j−1
(e
x, e
ϕℓ (e
x), De
ϕℓ (e
x), · · · , D j−1 e
ϕℓ (e
x)).
∂λ j−1
e by hypothAs the sequence (e
ϕℓ ) tends to e
ϕ in the space C j−1 (O, Z)
esis, we can apply Lemma A2.5 with j − 1 (which amount to shrinking
the neighbourhood O if necessary) and hence
||B j−1
x)|| ≤ β,
j−1,ℓ (e
for every e
xǫO. Inequality (A2.46) becomes
j−1
||D je
ϕℓ+1 (e
x) − D j e
ϕ(e
x)|| < ||bℓ (e
x) − b j−1 (e
x)||
+
j−1
X
i=0
j−1
j−1
j−1
||Biℓ (e
x) − Bi (e
x) − Bi (e
x)||||Di+1 e
ϕ||
(A2.47)
+
j−2
X
i=0
j−1
||Biℓ (e
x)||||Di+1 e
ϕℓ (e
x) − Di+1 e
ϕe
x||+
+ β||D je
ϕℓ − D j e
ϕ(e
x)||.
Now, since j ≤ k − 1, we can apply Lemma A2.6 (i) with j − 1: 271
There are constant q with 0 < q < 1 and C > 0 such that
j−1
||bℓ (e
x) − b j−1 (e
x)|| ≤ Cqℓ ,
j−1
j−1
||Biℓ (e
x) − Bi (e
x)|| ≤ Cqℓ , 0 ≤ i ≤ j − 1,
2. Complements of Chapter IV
214
e ⊂ C j (O, Z),
e there is a
for every ℓ ≥ 0 and every e
xǫO. As e
ϕǫC k (O, Z)
constant K > 0 such that
||Di+1 e
ϕ(e
x)|| ≤ K,
for every e
xǫO and 0 ≤ i ≤ j − 1. Together with (A2.47), these observations give
||D je
ϕℓ+1 (e
x) − D j e
ϕ(e
x)|| < ( jK + 1)Cqℓ
+
j−2
X
i=0
j−1
||Biℓ (e
x)||||Di+1 e
ϕℓ (e
x) − Di+1 e
ϕ(e
x)||
(A2.48)
j
j
+ β||D e
ϕℓ (e
x) − D e
ϕ(e
x)||.
From the convergence of the sequence (e
ϕℓ ) to e
ϕ geometrically in the
e it is not restrictive to assume that q with 0 < q < 1
space C j−1 (O, Z),
and C > 0 are such that
||Di+1 e
ϕℓ (e
x) − Di+1 e
ϕ(e
x)|| ≤ Cqℓ ,
for every e
xǫO, 0 ≤ i ≤ j − 2. Besides, we can suppose the constant K
large enough for the inequality
j−1
||Biℓ (e
x)|| ≤ K,
to hold for every e
xǫO, 0 ≤ i ≤ j − 2 and ℓ ≥ 0. Indeed, this follows from
the definition (A2.39), namely
j−1
Biℓ (e
x) =
272
∂Φ j−1
(e
x, e
ϕℓ (e
x), De
ϕℓ (e
x), · · · , D j−1 e
ϕℓ (e
x))
∂λi
and the compactness of the set O × Λ0 × · · · × Λ j−1 (Lemma A2.4 with
j − 1). Inequality (A2.48) then tales the simpler form
||D je
ϕℓ+1 (e
x) − D j e
ϕ(e
x)|| ≤ [(2 j − 1)K + 1]Cqℓ + β||D je
ϕℓ (e
x) − D j e
ϕ(e
x)||.
Replacing q by max(q, β) < 1 and setting
K0 =
1
[(2 j − 1)K + 1]C,
q
215
the above inequality becomes
||D j e
ϕℓ+1 (e
x) − D j e
ϕ(e
x)|| < K0 qℓ+1 + q||D j e
ϕℓ (e
x) − D je
ϕ(e
x)||
and yields
ϕℓ − e
ϕ)|∞,O .
|D j (e
ϕℓ+1 − e
ϕ)|∞,O ≤ K0 qℓ+1 + q|D j (e
(A2.49)
Let us define the sequence (aℓ )ℓ≥0 by



= |D je
ϕ|∞,O ,
a0


aℓ+1 = K0 qℓ+1 + qaℓ , ℓ ≥ 0
It is immediately checked that
|D j (e
ϕℓ − e
ϕ)|∞,O ≤ aℓ ,
(A2.50)
for every ℓ ≥ 0 (recall that e
ϕ0 = 0). On the other hand, the sequence
(aℓ ) is equivalently defined by
aℓ = (K0 ℓ + a0 )qℓ , ℓ ≥ 0.
Choosing q < q′ < 1, this is the same as
!
q
aℓ = (K0 ℓ + a0 ) ′ (q′ )ℓ , ℓ ≥ 0.
q
As 0 < q/q′ < 1, the term (K0 ℓ + a0 )(q/q′ )ℓ tends to 0 and hence is
uniformly bounded by a constant K1 > 0. Thus,
273
aℓ ≤ K1 (q′ )ℓ , ℓ ≥ 0
and our assertion follows from (A2.50).
Replacing k by k + 1 in Theorem A2.2, we get
Corollary A2.1 : Assume that the mapping Φ is of class C k+1 . Then,
after shrinking the neighbourhood O is necessary, the sequence (e
ϕℓ )
e
tends to the mapping e
ϕ geometrically in the space C k (O, Z).
2. Complements of Chapter IV
216
Oue nect theorem shows that even when the mapping Φ is only of
e but the rate
class C k , the sequence (e
ϕℓ ) tends to e
ϕ in the space C k (O, Z)
of convergence is not known in general (compare with Corollary A2.1
above).
Theorem A2.3: After shrinking the neighbourhood O if necessary, the
e
sequence (e
ϕℓ ) tends to the mapping e
ϕ in the space C k (O, Z).
Proof. From Theorem A2.2, we know that the sequence (e
ϕℓ ) tends to e
ϕ
k−1
e
in the space C (O, Z) and it remains to prove the convergence of the
e To do this,
sequence (Dk e
ϕℓ )ℓ≥0 to Dk e
ϕ in the space C 0 (O, Lk (Rn+1 , Z)).
we can repeat a part of the proof of Theorem A2.2. Replacing j by k,
the observations we made up to and including formula (A2.47) are still
valid and we then find
x) − bk−1 (e
x)||
||Dk e
ϕℓ+1 (e
x) − Dk e
ϕ(e
x)|| ≤ ||bk−1
ℓ (e
+
k−1
X
+
k−2
X
i=0
i=0
274
||Bk−1
x) − Bk−1
x)||||Di+1 e
ϕ(e
x)|| (A2.51)
i (e
iℓ (e
||Bk−1
x)||||Di+1 e
ϕℓ (e
x) − Di+1 e
ϕ(e
x)||
iℓ (e
+ β||Dk e
ϕℓ (e
x ) − Dk e
ϕ(e
x)||,
for every e
x ∈ O and every ℓ ≥ 0. Here, we can no longer use Lemma
x) − bk−1 (e
x)|| or
A2.6 (i) for finding an estimate of the terms ||bk−1
ℓ (e
k−1
k−1
x) − Bi (e
x)|| but Lemma A2.6 (ii) applies with j = k − 1: Given
||Biℓ (e
ǫ > 0 and ℓ large enough, say ℓ ≥ ℓ0 , one has
x) − bk−1 (e
x)|| ≤ ǫ,
||bk−1
ℓ (e
||Bk−1
x) − Bk−1 (e
x)|| ≤ ǫ, 0 ≤ i ≤ k − 1,
iℓ (e
for every e
x ∈ O. By the same arguments as in Theorem A2.2 and from
e we get the existence of a constant K > 0
the regularity e
ϕ ∈ C k (O, Z),
such that
||Di+1 e
ϕ(e
x)|| ≤ K,
217
for every e
x ∈ O ≤ i ≤ k − 1 and
||Bk−1
x)|| ≤ K,
iℓ (e
for every e
x ∈ O, 0 ≤ i ≤ k − 2 and ℓ ≥ 0 and ℓ0 can be taken large
enough for the inequality
||Di+1 e
ϕℓ (e
x) − Di+1 e
ϕ(e
x)|| ≤ ǫ
to hold for every e
x ∈ O, 0 ≤ i ≤ k − 2. Together with (A2.51), we arrive
at
||Dk e
ϕℓ+1 (e
x ) − Dk e
ϕ(e
x)|| ≤ [(2k − 1)K + 1]ǫ + β||Dk e
ϕℓ (e
x) − Dk e
ϕ(e
x)||.
Setting
K0 = (2k − 1)K + 1,
the above inequality becomes
and yields
||Dk e
ϕℓ+1 (e
x ) − Dk e
ϕ(e
x)|| ≤ K0 ǫβ||Dk e
ϕℓ (e
x) − Dk e
ϕ(e
x)||
275
|Dk (e
ϕℓ+1 − e
ϕ)|∞,O < K0 ǫ + β|Dk (e
ϕℓ − e
ϕ)|∞,O .
Let us define the sequence (aℓ )ℓ≥ℓ0 by



ϕℓ0 − e
ϕ)|∞,O ,
 aℓ0 = |Dk (e


 aℓ+1 = K0 ǫ + βaℓ , ℓ ≥ ℓ0 .
It is immediately checked that
|Dk (e
ϕℓ − e
ϕ)|∞,O ≤ aℓ ,
(A2.52)
for every ℓ ≥ ℓ0 . On the other hand, the sequence (aℓ )ℓ≥ℓ0 is equivalently
defined by
aℓ = K0
1 − βℓ−ℓ0
ǫ + βℓ−ℓ0 aℓ0 , ℓ ≥ ℓ0 + 1.
1−β
2. Complements of Chapter IV
218
For ℓ ≥ ℓ0 large enough, say ℓ ≥ ℓ1 , one has βℓ−ℓ0 aℓ0 ≤ ǫ.
Thus
"
#
K0
aℓ ≤
+ 1 ǫ,
1−β
for ℓ ≥ ℓ1 and our assertion follows from (A2.52).
Here is now a result of a different kind that we shall use later on.
Lemma A2.7: For every ℓ ≥ 0, one has
for 0 ≤ i ≤ k.
276
Di e
ϕℓ+k (0) = Die
ϕ(0),
(A2.53)
Proof. Again, we shall proceed by induction and show for 0 ≤ j ≤ k
and every ℓ ≥ 0
Di e
ϕℓ+ j (0) = Die
ϕ(0),
(A2.54)
for 0 ≤ i ≤ j. This result is true for j = 0 since e
ϕ(0) = 0 and e
ϕℓ (0) = 0
as it follows from the definition. Assume then that j ≥ 1 and the result
is true up to ranke j − 1, namely, that
Di e
ϕℓ+ j−1 (0) = Die
ϕ(0),
(A2.55)
for 0 ≤ i ≤ j − 1. From the relations (A2.29) and (A2.30)
Di e
ϕℓ+ j (0) = Φi (0, e
ϕℓ+ j−1 (0), · · · , Die
ϕℓ+ j−1 (0)),
i
i
De
ϕ(0) = Φi (0, e
ϕ(0), · · · , D e
ϕ(0)),
(A2.56)
(A2.57)
for 0 ≤ i ≤ k and hence for 0 ≤ i ≤ j. Clearly, from (A2.55), the relation
(A2.54) holds for 0 ≤ i ≤ j − 1 but we must show that it also holds for
i = j.
The fact that e
ϕ(0) = e
ϕℓ+ j−1 (0) = 0 is essentail here because taking
i = j in (A2.56) and (A2.57) and coming back to the definition (cf.
(A2.28)), we get
D je
ϕℓ+ j (0) =
∂Φ j−1
(0, 0, De
ϕℓ+ j−1 (0), · · · , D j−1 e
ϕℓ+ j− (0)) +
∂e
x
(A2.58)
219
+
j−1
X
∂Φ j−1
∂λi
i=0
and
(0, 0, De
ϕℓ+ j−1 (0), · · · , D j−1 e
ϕℓ+ j−1 (0))Di+1 e
ϕℓ+ j−1 (0)
D je
ϕ(0) =
+
∂Φ j−1
(0, 0, De
ϕ(0), · · · , D j−1 e
ϕ(0)) +
∂e
x
j−1
X
∂Φ j−1
i=0
∂λi
(A2.59)
(0, 0, De
ϕ(0), · · · , D j−1 e
ϕ(0))Di+1 e
ϕ(0).
But, in view of Lemma A2.3, the term corresponding with i = 277
j − 1 vanishes in both expressions (A2.58) and (A2.59). It follows
that D je
ϕℓ+ j (0) and D je
ϕ(0) are given by the same formula involving the
derivatives of order ≤ j − 1 of the mappings e
ϕℓ+ j−1 and e
ϕ respectively.
As these derivatives coincide by the hypothesis of induction, relation
(A2.54) at rank j is established and the proof is complete.
let us now consider the mapping FǫC k (O × B(0, ρ), Rn ) introduced
in §6 of Chapter 4 verifying F(0) = 0. So as to fully prove the results
we used in §6 of Chapter 4, we still have to show that the sequence of
mapping
e
xǫO → fℓ (e
x) = F(e
x, e
ϕℓ+k (e
x)),
of class C k from O to Rn tends to the mapping
e
xǫO → f (e
x) = F(e
x, e
ϕ(e
x)),
in the space C k (O, Rn ), the convergence being geometrical in the space
C k−1 (O, Rn ) (and also geometrical in the space C k (O, Rn ) if both mappings Φ and F are of class C k+1 ). Besides, we used the relation
D j fℓ (0) = D j f (0), 0 ≤ j ≤ k,
which is now immediate from Lemma A2.7 because, for a given index
0 ≤ j ≤ k, the derivatives of order j of the mappings fℓ and f are
given by the same formula involving the derivatives of order ≤ j of the
mappings F and e
ϕℓ+k on the one hand those of F and e
ϕ on the other
hand.
2. Complements of Chapter IV
220
e and setting 278
Still denoting by λ0 the generic element of the space Z
F0 = F, define for 1 ≤ j ≤ k the mappings
F j : O × B(0, ρ) ×
j
Y
i=1
e → L j (Rn+1 , Rn ),
Li (Rn+1 , Z)
by
∂F j−1
(e
x, λ0 , · · · , λ j−1 ) +
∂e
x
j−1
X
∂F j−1
+
(e
x, λ0 , · · · , λ j−1 )λi+1 .
∂λi
i=0
F j (e
x, λ0 , · · · , λ j ) =
(A2.60)
Of course, the mappping F j play with F the role that the mappings
Φ j play with Φ and have similar properties. For instance, the mapping
F j is of class C k− j and for e
xǫO,
D j fℓ (e
x) = F j (e
x, e
ϕℓ+k (e
x), De
ϕℓ+k (e
x), · · · , D je
ϕℓ+k (e
x))
(A2.61)
D j f (e
x) = F j (e
x, e
ϕ(e
x), De
ϕ(e
x), · · · , D je
ϕ(e
x)).
(A2.62)
for every ℓ ≥ 0, while
For 0 ≤ j ≤ k − 1 and e
xǫO, we shall set
∂F j
(e
x, e
ϕ(e
x), De
ϕ(e
x), · · · , D je
ϕ(e
x)),
(A2.63)
∂e
x
∂F j
j
cℓ (e
x) =
(e
x, e
ϕℓ+k (e
x), De
ϕℓ+k (e
x), · · · , D je
ϕℓ+k (e
x)) for ℓ ≥ 0. (A2.64)
∂e
x
c j (e
x) =
Similarly, for 0 ≤ j ≤ k − 1, 0 ≤ i ≤ j and e
xǫO, set
∂F j
(e
x, e
ϕ(e
x), De
ϕ(e
x), · · · , Die
ϕ(e
x)),
(A2.65)
∂λi
∂F j
j
Ciℓ (e
x) =
(e
x, e
ϕℓ+k (e
x), De
ϕℓ+k (e
x), · · · , Die
ϕℓ+k (e
x)) for ℓ ≥ 0 (A2.66)
∂λ j
j
Ci (e
x) =
In what follows, we assume that the neighbourhood O has been chosen so that Theorem A2.2 and A2.3 apply
221
279
Lemma A2.8: (i) Assume k ≥ 2 and 0 ≤ j ≤ k − 2. Then, the sequence
j
(cℓ )ℓ≥0 tends to c j geometrically in the space C 0 (O, L (Rn+1 , L j (Rn+1 ,
j
j
Rn ))) and the sequence (Ciℓ )ℓ≥0 tends to Ci in the space C 0 (O,
n+1
n
n+1
n
L (Li (R , R ), L j (R , R ))).
j
(ii) Assume only k ≥ 1 and 0 ≤ j ≤ k − 1. Then, the sequence (cℓ )ℓ≥0
tends to c j in the space C 0 (O, L (Rn+1 , L j (Rn+1 , Rn ))) and the sequence
j
j
(Ciℓ )ℓ≥0 tends to Ci in the space C 0 (O, L (Li (Rn+1 , Rn ), Li (Rn+1 , Rn ))).
Proof. The proof of this lemma parallels that of Lemma A2.6. Note
only that the (geometrical) convergence of the sequence (e
ϕℓ+k ) to e
ϕ in
j
e
the space C (O, Z) need not be listed in the assumption since it is known
from Theorem A2.2
Theorem A2.4 : The sequence ( fℓ ) tends to f in the space C k (O, Rn )
and the convergence is geometrical in the space C k−1 (O, Rn ).
Proof. Once again, it is enough to prove equivalently that the sequence
(D j fℓ )ℓ≥0 tends to D j f in the space C 0 (O, L j (Rn+1 , Rn )) for 0 ≤ j ≤
k, the convergence being geometrical for 0 ≤ j ≤ k − 1. We shall
distinguish the case j = 0 and j ≥ 1.
When j = 0, we must prove that the sequence ( fℓ ) tends to f geometrically in the space C 0 (O, Rn ). By definition and with the Taylor
formula (since F is at least C 1 ), one has for e
xǫO and ℓ ≥ 0
fℓ (e
x) − f (e
x) = F(e
x, e
ϕℓ+k (e
x)) − F(e
x, e
ϕ(e
x))
Z 1
∂F
=
(e
x, e
ϕ(e
x) + s(e
ϕℓ+k (e
x) − e
ϕ(e
x))) · (e
ϕℓ+k (e
x) − e
ϕ(e
x))ds.
0 ∂λ0
By the continuity of the mapping (∂F/∂λ0 ) on the compact set O × 280
where Λc0 ⊂ B(0, ρ) denotes the closed convex hull of the compact
set Λ0 (cf. Lemma A2.4), there is a constant K > 0 such that
Λc0 ,
||
∂F
(e
x, e
ϕ(e
x) + s(e
ϕℓ+k (e
x) − e
ϕ(e
x)))|| ≤ K,
∂Λ0
for every e
xǫO and ℓ ≥ 0. Hence
| fℓ − f |∞,O ≤ K|e
ϕℓ+k − e
ϕ|∞,O
2. Complements of Chapter IV
222
and the geometrical rate off convergence of the sequence ( fℓ ) to f in the
space C 0 (O, Rn ) follows from Theorem A2.2.
When 1 ≤ j ≤ k and with (A2.60) - (A2.62) and the notation
(A2.63)-(A2.66), the method we used in Theorem A2.2 and A2.3 leads
to the inequality
j−1
||D j fℓ (e
x) − D j f (e
x)|| ≤ ||cℓ (e
x) − C j−1 (e
x)||
+ ||
+
j−1
X
i=0
j−1
X
i=0
j−1
j−1
||ciℓ (e
x) − ||Ci
(e
x)||||Di+1 ||e
ϕ(e
x)||
||Ciℓj−1 (e
x)||||||Di+1 e
ϕℓ+k (e
x) − ||Di+1 e
ϕ(e
x)||,
for every e
xǫO. Due to the compactness of the set O × Λ0 × · · · × Λ j−1
e there is a constant K > 0
(Lemma A2.4) and the regularity e
ϕǫC k (O, Z),
such that
||Di+1 e
ϕ(e
x)|| ≤ K,
||C − ıℓ j−1 (e
x)|| ≤ K,
for e
xǫO, 0 ≤ i ≤ j − 1 and ℓ ≥ 0. Thus, we first obtain
j−1
||D j fℓ (e
x) − D j f (e
x)|| < ||cℓ (e
x) − c j−1 (e
x)
 j−1

X j−1

j−1
i+1
i+1

+ K  ||Ciℓ (e
x) − Ci (e
x)|| + ||D e
ϕℓ+k (e
x) − D e
ϕ(e
x)||
i=0
281
and next
|D j ( fℓ − f )|∞,O ≤ |cℓj − c j−1 |∞,O
 j−1

X j−1

j−1
i+1
+ K|  |Ciℓ − Ci |∞,O + D (e
ϕℓ+k − e
ϕ)|∞,|O  .
i=0
(A2.67)
For k ≥ 2 and 1 ≤ j ≤ k − 1 (i.e. 0 ≤ j − 1 ≤ k − 2). this inequality
yields the geometrical convergence of the sequence (D j fℓ )ℓ≥0 to D j f in
223
the space C 0 (O, L j (Rn+1 , Rn )) by applying Theorem A2.2 and Lemma
A2.8 (i). For k ≥ 1 and j = k, convergence of the sequence (Dk fℓ )ℓ≥0
to Dk f (with no rate of convergence in the space C 0 (O, Lk (Rn+1 , Rn ))
also follows from (A2.67) by applying Theorem A2.3 and Lemma A2.8
(ii).
Corollary A2.2: Assume that both mappings F and Φ are of class C k+1 .
Then, the sequence ( fℓ ) tends to the maping f geometrically in the space
C k (O, Rn ).
Proof. When the mapping F is of class C k−1 , the mappings F j 0 ≤ j ≤
j
j
j
k, are of class C k+1− j and hence the mappings c j , cℓ , Ci and Ciℓ are of
1
class C at least. This allows us to prove Lemma A2.8 (i) for the indices
0 ≤ j ≤ k − 1 (instead of 0 ≤ j ≤ k − 2). On the other hand, if the
mapping Φ is of class C k+1 too, the sequence (e
ϕℓ ) tends to the mapping
k
e
e
ϕ geometrically in the space C (O, Z) (Corollary A2.1) and so does the
sequence (e
ϕℓ+k ). The result follows now from inequality (A2.67) for
1 ≤ j ≤ k (i.e. 0 ≤ j − 1 ≤ k − 1), the geometrical convergence of the
sequence ( fℓ ) to f in the space C 0 (O, Rn ) having been already proved in
Theorem A2.3.
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